>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE   1

                             USER'S MANUAL FOR

       *************************************************************
       *  MICROSCOPE: A SOFTWARE SYSTEM FOR MULTIVARIATE ANALYSIS  *
       *************************************************************

                                     BY

                        PETER ALFELD AND BILL HARRIS

                         DEPARTMENT OF MATHEMATICS
                             UNIVERSITY OF UTAH
                               SALT LAKE CITY
                                 UTAH 84112

                     TEL.: 801-581-6842 OR 801-581-6851

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                                  ABSTRACT

          THIS  MANUAL  DESCRIBES  MICROSCOPE:  A PORTABLE FORTRAN SOFTWARE
SYSTEM FOR THE ANALYSIS OF MULTIVARIATE FUNCTIONS.  GIVEN AN  INTERPOLATION
OR  APPROXIMATION  SCHEME, IT ALLOWS THE FOLLOWING QUESTIONS, AMONG OTHERS,
TO   BE  ANSWERED:    DOES  THE  SCHEME  INTERPOLATE?    HOW  OFTEN  IS  IT
DIFFERENTIABLE?    WHAT FUNCTIONS DOES IT REPRODUCE EXACTLY?  IF THE SCHEME
IS POLYNOMIAL, WHAT IS ITS POLYNOMIAL DEGREE?  WHERE IS THE SMOOTHNESS OF A
FUNCTION REDUCED?  WHERE ARE THE BUGS IN A FORTRAN IMPLEMENTATION?

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                              LIST OF CONTENTS

   0. ABOUT THIS MANUAL .......................................6

   1. INTRODUCTION ............................................8
  1.1 MULTIVARIATE INTERPOLATION AND APPROXIMATION ............8
  1.2 THE BASIC APPROACH ......................................9
  1.3 THE ALPHANUMERICAL DISPLAY ..............................13
  1.4 SUMMARY .................................................19

   2. APPLICATIONS ............................................21
  2.1 INTERPOLATION ...........................................22
  2.2 SMOOTHNESS ..............................................27
  2.3 DEGREE OF PRECISION .....................................28
  2.4 DEGREE OF A POLYNOMIAL ..................................30

   3. USING MICROSCOPE ........................................33
  3.1 THE MCRSCP ROUTINE ......................................34
  3.2 THE COMMAND MODE ........................................36
  3.3 A DISCONTINUITY IN THE SECOND DERIVATIVE ................38

   4. A DETAILED DESCRIPTION OF FEATURES AND COMMANDS .........44
  4.1 ROUND-OFF EFFECTS .......................................44
  4.2 DERIVATIVES OF STEP FUNCTIONS ...........................45
  4.3 THE WINDOW ..............................................49
  4.4 REFERENCE TABLE OF COMMANDS .............................51
  4.5 COMMANDS TO CONTROL THE DISPLAY .........................54
  4.6 COMMANDS TO CONTROL THE POINT OF EXAMINATION ............58
  4.7 COMMANDS TO CONTROL THE DIRECTION OF INVESTIGATION ......59
  4.8 COMMANDS TO CONTROL ACCURACY, THE WINDOW,
          AND ROUND-OFF EFFECTS ...............................60
  4.9 COMMANDS TO CONTROL EXECUTION ...........................62
 4.10 COMMANDS TO OBTAIN ON-LINE HELP .........................64
 4.11 COMMANDS TO KEEP A RECORD ...............................65
 4.12 COMMANDS TO ANALYZE CROSS DERIVATIVES ...................65
 4.13 PROGRAMMING MICROSCOPE ..................................67
 4.14 THE PLOT COMMAND FOR OBTAINING A <PLOT79> DISPLAY .......69
 4.15 USER INTERVENTION .......................................73
 4.16 DEFAULT SETTINGS ........................................74

   5. INSTALLATION GUIDE ......................................75
  5.1 DESCRIPTION OF THE PACKAGE ..............................75
  5.2 A KEY TO SELECTING THE APPROPRIATE FILES ................80

  APPENDIX I: NUMERICAL DIFFERENTIATION .......................82
           I.1 THE ROUND-OFF THRESHOLD ........................85
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           I.2 ROUND-OFF VERSUS TRUNCATION ERRORS .............87
           I.3 DERIVATIVES OF STEP FUNCTIONS ..................89
  APPENDIX II: ORGANIZATION OF MICROSCOPE .....................92
           II.1 ORGANIZATION OF THE PROGRAM PACKAGE ...........92
           II.2 SAMPLING SCHEME AND LOGIC .....................95
           II.3 COMMON BLOCKS .................................104
  APPENDIX III: THE TEST PACKAGE ..............................110
  APPENDIX IV: EXAMPLES FOR GRAPHICAL (<PLOT79>) DISPLAYS .....115
  REFERENCES ..................................................116
  ACKNOWLEDGEMENTS ............................................117
  INDEX .......................................................118

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                         LIST OF TABLES AND FIGURES

   FIGURE 1 THE ALPHANUMERICAL DISPLAY ........................14
   FIGURE 2 INTERPOLATION TO POSITION AT ZERO..................22
   FIGURE 3 ...................................................23
   FIGURE 4 INTERPOLATION AT X = 2 ............................24
   FIGURE 5 INTERPOLATION TO THE FIRST DERIVATIVE .............25
   FIGURE 6 ...................................................26
   FIGURE 7 FREE END CONDITION ................................27
   FIGURE 8 SMOOTHNESS AT X = 1 ...............................28
   FIGURE 9 A ZERO ERROR ......................................29
   FIGURE 10 ROUND-OFF WITH A BIAS ............................30
   FIGURE 11 A CUBIC FUNCTION .................................31
   FIGURE 12 ..................................................32
   FIGURE 13 ..................................................38
   FIGURE 14 ..................................................39
   FIGURE 15 ..................................................40
   FIGURE 16 ..................................................41
   FIGURE 17 ..................................................42
   FIGURE 18 AT THE LIMITS OF RESOLUTION ......................43
   FIGURE 19 DERIVATIVES OF A STEP FUNCTION ...................46
                               0
   FIGURE 20 DERIVATIVES OF A C  FUNCTION .....................47
                               1
   FIGURE 21 DERIVATIVES OF A C  FUNCTION .....................47
                               2
   FIGURE 22 DERIVATIVES OF A C  FUNCTION .....................48
                               3
   FIGURE 23 DERIVATIVES OF A C  FUNCTION .....................48
                               4
   FIGURE 24 DERIVATIVES OF A C  FUNCTION .....................49
   FIGURE 25 AN INVISIBLE GRAPH ...............................56
   FIGURE 26 THE DISENTANGLED GRAPHS ..........................57
   FIGURE 27 AN EXTENDED DISCONTINUITY ........................67
   FIGURE 28 A 3/2 TIMES DIFFERENTIABLE FUNCTION ..............113
   FIGURE 29 ANOTHER 3/2 TIMES DIFFERENTIABLE FUNCTION ........114

   TABLE: EFFICIENCY OF WINDOW WIDTHS .........................50
   QUICK REFERENCE TABLE OF COMMANDS ..........................52
   TABLE OF DISCRETIZATION CONTROLING COMMANDS ................60
   TABLE: RELATIVE EFFORTS ....................................62
   TABLE 1: CONSTANTS FOR DIFFERENTIATION FORMULAS ............84
   TABLE 2: THRESHOLD VALUES OF H .............................86
   TABLE 3: OPTIMUM VALUES OF H ...............................88
   TABLE 4: THE SUPERIORITY OF TESTING HIGHER DERIVATIVES .....91
   EPITOME FLOWCHART ..........................................94
   OUTLINE OF SGAMMA ..........................................104
   TABLE OF TEST PARAMETERS ...................................112

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                            0. ABOUT THIS MANUAL

          THIS    MANUAL   CONSTITUTES  THE  COMPREHENSIVE  DESCRIPTION  OF
MICROSCOPE, ITS OPERATION, APPLICATIONS, FOUNDATIONS, AND INNER WORKINGS.

          SECTION 1.1 DESCRIBES WHAT KIND OF PROBLEMS CAN BE  ADDRESSED  BY
MICROSCOPE.    SECTION 1.2 DESCRIBES THE BASIC APPROACH AND IS FUNDAMENTAL.
SECTION 1.3 DESCRIBES THE ALPHANUMERICAL SCREEN DISPLAY AND IS  NEEDED  FOR
UNDERSTANDING  THE  EXAMPLES  GIVEN  THROUGHOUT  THIS  MANUAL.    IT   ALSO
INTRODUCES SOME  TERMS  AND  CONCEPTS  AND  SERVES  AS  A  PRELUDE  TO  THE
CAPABILITIES OF MICROSCOPE.

          SECTION 2 GIVES EXAMPLES THAT ILLUSTRATE FOUR MAJOR  APPLICATIONS
(INTERPOLATION,  SMOOTHNESS,  DEGREE  OF PRECISION, AND POLYNOMIAL DEGREE).
THESE DO OF COURSE NOT EXHAUST THE POTENTIAL OF MICROSCOPE.  IN SECTION  3,
ONE  OTHER  APPLICATION  (DISCOVERY  OR  IDENTIFICATION  OF  A  POINT WHERE
SMOOTHNESS IS REDUCED) IS USED AS A VEHICLE FOR INTRODUCING  THE  MECHANICS
OF USING MICROSCOPE.

          SECTION 4 GIVES  A  DETAILED  DESCRIPTION  OF  ALL  COMMANDS  AND
FEATURES,  ORGANIZED  BY  TYPE  OF  FEATURE OR TASK TO BE ACCOMPLISHED.  IN
REGULAR USE, THIS WILL PROBABLY BE THE MOST FREQUENTLY CONSULTED SECTION.

          IN  READING  SECTIONS  2  AND  3,  IT IS DESIRABLE THAT A WORKING
VERSION OF MICROSCOPE IS AVAILABLE SO  THAT  HANDS  ON  EXPERIENCE  CAN  BE
GAINED.   IF THIS IS NOT THE CASE THE INSTALLATION GUIDE (SECTION 5) SHOULD
BE CONSULTED FIRST.  THAT SECTION ALSO CONTAINS A  DETAILED  DISCUSSION  OF
THE PORTABILITY FEATURES AND POSSIBLE VARIANTS OF MICROSCOPE.

          APPENDICES ARE INCLUDED THAT DESCRIBE SOME TECHNICAL FEATURES  IN
DETAIL:    THE  DIFFERENTIATION  FORMULAS,  THE ORGANIZATION OF THE PROGRAM
PACKAGE, AND THE TEST PACKAGE.    THERE  SHOULD  ALSO  BE  AN  APPENDIX  IV
CONTAINING PEN PLOTS CORRESPONDING TO THE CRUDER ALPHANUMERICAL GRAPHICS IN
THE BODY OF THE MANUAL.  HOWEVER, THAT APPENDIX IS NOT MACHINE LEGIBLE, AND
IS DISTRIBUTED SEPARATELY.

          THE MANUAL ITSELF IS IN MACHINE READABLE FORM SO THAT IT  CAN  BE
DISTRIBUTED  TOGETHER WITH THE PACKAGE AND CAN BE PRINTED LOCALLY, AND READ
ON LINE USING FOR EXAMPLE THE SEARCH FEATURES OF A TEXT  EDITOR.    IT  WAS
GENERATED  BY  A  MICROSCOPE  PROGRAM,  WITH  THE  TEXT OF THE MANUAL BEING
"NOTES" AND THE EXAMPLE SCREEN DISPLAYS BEING COMPUTED  BY  MICROSCOPE  AND
AUTOMATICALLY  EMBEDDED  INTO  THE TEXT.  (ACTUALLY, THE MICROSCOPE PROGRAM
GENERATED A DATA FILE THAT WAS THEN FED INTO THE  TEXT  FORMATTING  PROGRAM
DOCUMENT,  SEE  BEEBE, 1980.) MACHINE LEGIBILITY, HOWEVER, ACCOUNTS FOR THE
SOMEWHAT AWKWARD NOTATION IN SOME PLACES.  FOR EXAMPLE, GREEK LETTERS  HAVE
BEEN GIVEN ABBREVIATIONS IN TERMS OF ROMAN LETTERS.

          DEFINITIONS ARE DENOTED BY WRITING THE  DEFINED  TERM  IN  SINGLE

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QUOTATION MARKS.  ORDINARY QUOTATIONS ARE IN DOUBLE QUOTATION MARKS.

          THE EXAMPLES IN THIS MANUAL ARE REPRODUCIBLE  SINCE  THE  SET  OF
ROUTINES  USED FOR THEIR GENERATION IS SUPPLIED WITH THE PACKAGE AND CAN BE
USED FOR TEST PURPOSES.   TO  ENSURE  UNIFORMITY  OF  RESULTS  OBTAINED  ON
DIFFERENT MACHINES, A FACILITY HAS BEEN ADDED TO SIMULATE ROUNDING, AND ALL
EXAMPLES HAVE BEEN COMPUTED IN 10 DIGITS ARITHMETIC.  MORE PRECISELY, IF  X
IS THE QUANTITY TO BE ROUNDED TO D DIGITS, SAY, THEN X IS REPLACED BY

                                     -D             -D
                      Z := (1+EPS *10  )*X + EPS *10
                                 1              2

WHERE EPS  AND EPS  ARE RANDOM NUMBERS BETWEEN -1 AND +1.  THE ADDITION  OF
         1        2
THE  EPS   TERM  IS  NOT  STANDARD  BUT APPROPRIATE IN THE PRESENT CONTEXT,
        2
BECAUSE IN INVESTIGATIONS WITH MICROSCOPE SMALL NUMBERS ARE  OFTEN  DUE  TO
TAKING  DIFFERENCES  BETWEEN  VERY  CLOSE  LARGE  NUMBERS,  LEADING  TO   A
CANCELLATION OF SIGNIFICANT DIGITS.  THE LARGER THE  MAGNITUDE  OF  X,  THE
MORE  INSIGNIFICANT  THE  SECOND  TERM  WILL  BECOME.  IF X IS ROUGHLY 1 OR
LARGER (IN  MAGNITUDE)  OUR  ROUNDING  IS  ESSENTIALLY  EQUIVALENT  TO  THE
STANDARD ROUNDING (WHERE EPS  = 0).
                            2

          THIS MANUAL WILL RARELY BE READ SEQUENTIALLY FROM FRONT TO  BACK.
THEREFORE,  KEY IDEAS AND CONCEPTS HAVE SOMETIMES BEEN RESTATED IF THEY ARE
CRUCIAL IN THE GIVEN CONTEXT.   IT  IS  HOPED  THAT  THIS  REDUNDANCY  WILL
CONTRIBUTE TO THE USEFULNESS OF THIS DOCUMENT.

          ANY CRTICISMS, COMMENTS  AND  SUGGESTIONS  ABOUT  MICROSCOPE  ARE
WELCOME  AND SHOULD BE DIRECTED TO PETER ALFELD, DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF UTAH, SALT LAKE CITY,  UTAH  84112,  801-581-6842.    WE  ARE
PARTICULARLY  INTERESTED  IN  ANY  APPLICATIONS  OF MICROSCOPE THAT ARE NOT
DESCRIBED IN THIS MANUAL.

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                              1. INTRODUCTION

             1.1  MULTIVARIATE INTERPOLATION AND APPROXIMATION

          THIS  MANUAL  DESCRIBES  A  FORTRAN SOFTWARE PACKAGE, MICROSCOPE,
WHOSE MAIN APPLICATIONS  ARE  IN  THE  ANALYSIS  OF  FUNCTIONS  OF  SEVERAL
VARIABLES  (OR  'SURFACES')  AS  THEY  OCCUR  E.G.  IN THE APPROXIMATION OF
MULTIVARIATE DATA OR IN THE DESIGN OF GEOMETRIC OBJECTS SUCH AS THE BODY OF
AN  AUTOMOBILE  OR  AN  AIRCRAFT.  THIS IS A CURRENTLY VERY ACTIVE RESEARCH
AREA WITH MANY UNSOLVED AND DIFFICULT PROBLEMS.  FOR A SURVEY SEE BARNHILL,
1983, AND THE REFERENCES QUOTED THEREIN.

          MICROSCOPE IS PORTABLE TO THE EXTENT THAT IT HAS PASSED THE PFORT
VERIFIER  (RYDER,  1974)  WITH  THE EXCEPTION THAT IT USES SOME NON-FORTRAN
CHARACTERS (NAMELY ?,<,>,!,:,;).  SOME NON-PORTABLE OPTIONAL  FEATURES  ARE
SUPPORTED  BY  VARIANTS  OF  THE CODE.  THESE INCLUDE THE USE OF LOWER CASE
LETTERS   IN  INPUT  AND  OUTPUT,  AND  THE  USE  OF  MODIFICATIONS  OF  AN
ALPHANUMERICAL DISPLAY ON A CRT TERMINAL.  IF THE SOFTWARE (<PLOT79>, BEEBE
1979) AND HARDWARE ARE AVAILABLE MORE SOPHISTICATED AND PLEASING  GRAPHICAL
DISPLAYS  CAN  ALSO  BE  OBTAINED.  HOWEVER, EVEN IF NONE OF THE ADDITIONAL
FEATURES ARE PROVIDED BY A PARTICULAR INSTALLATION, A WORKING, IF  SOMEWHAT
CRUDE,  VERSION  OF  MICROSCOPE  CAN  BE CONSTRUCTED.  SEE THE INSTALLATION
GUIDE (SECTION 5) FOR DETAILS.

          THE  OBJECTS  THAT CAN BE EXAMINED BY MICROSCOPE ARE FUNCTIONS OF
FROM ONE TO THREE  INDEPENDENT  VARIABLES.    ALLOWING  ONLY  UP  TO  THREE
VARIABLES ENABLED US TO CONTAIN ALL NUMERICAL INFORMATION ON THE CRT SCREEN
AND PROVED SUFFICENT FOR ALL CASES WE HAVE ENCOUNTERED.  IF A  FUNCTION  OF
MORE  THAN  THREE VARIABLES MUST BE EXAMINED MICROSCOPE CAN BE APPLIED TO A
SUITABLE RESTRICTION OF THE FUNCTION TO A SUBDOMAIN OF  DIMENSION  AT  MOST
THREE.    IF  THERE  IS  SUFFICENT  INTEREST, FUTURE VERSIONS OF MICROSCOPE
CAPABLE OF HANDLING MORE THAN THREE VARIABLES WILL BE PROVIDED.

          THE  FUNCTIONS  OF  INTEREST  ARE USUALLY OBTAINED BY APPLYING AN
INTERPOLATION OR APPROXIMATION SCHEME  TO  DISCRETE  OR  TRANSFINITE  (I.E.
INFINITELY  MANY)  DATA.    AS FAR AS THE USE OF MICROSCOPE IS CONCERNED IT
MATTERS LITTLE IF A FUNCTION IS OBTAINED BY INTERPOLATION OR  APPROXIMATION
(EXCEPT  THAT  ONE  WILL  NOT  NEED TO TEST FOR INTERPOLATION IN THE LATTER
CASE), SO WE WILL USE THE TWO TERMS INTERCHANGEABLY.

          WE  WILL  REFER  TO  THE  FUNCTION  THAT IS BEING EXAMINED AS THE
'TRIAL    FUNCTION'.    THE  TRIAL  FUNCTION  MAY  BE  DIFFERENT  FROM  THE
INTERPOLATING  OR  APPROXIMATING  FUNCTION  IN  WHICH  THE  FINAL USER OF A
NUMERICAL SCHEME IS INTERESTED.    FOR  EXAMPLE,  IT  IS  OFTEN  USEFUL  TO
GENERATE  DATA  FROM  A  'PRIMITIVE  FUNCTION'  AND THEN USE THE DIFFERENCE
BETWEEN THE PRIMITIVE FUNCTION AND ITS INTERPOLANT AS THE  TRIAL  FUNCTION.

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THIS  MAKES DISCONTINUITIES IN DERIVATIVES LARGE RELATIVE TO THE DERIVATIVE
VALUE AND THEREBY AMPLIFIES DISCONTINUITIES  WHICH  ARE  THEN  MORE  EASILY
DETECTED.

          FOUR TYPES OF QUESTIONS OCCUR MOST FREQUENTLY IN THE ANALYSIS  OF
THE TRIAL FUNCTION:

          -1-   DOES THE SCHEME INTERPOLATE?

          -2-   HOW  OFTEN  IS  THE  TRIAL  FUNCTION DIFFERENTIABLE?  (I.E.
WHAT IS ITS 'DEGREE OF SMOOTHNESS'?)

NOTE:THROUGHOUT  ALMOST  ALL OF THIS MANUAL WE WILL CONSIDER ONLY FUNCTIONS
     THAT POSSESS A STEP DISCONTINUITY IN SOME DERIVATIVE, FOR THIS IS  THE
     TYPE  OF  SMOOTHNESS  LIMITATION  THAT    OCCURS    IN    MULTIVARIATE
     INTERPOLATION    AND    APPROXIMATION.   THERE  ARE  OF  COURSE  OTHER
                                                             4/3
     POSSIBILITIES.  FOR  EXAMPLE,  THE  FUNCTION  F(X)  =  X     IS   ONCE
     DIFFERENTIABLE BUT ITS SECOND DERIVATIVE EXHIBITS A POLE RATHER THAN A
     STEP.  THE  TEST  PACKAGE  (SEE APPENDIX III) CAN BE USED TO GAIN SOME
     EXPERIENCE  WITH  SUCH  FUNCTIONS.  IN  PARTICULAR,  THE  EXAMPLES  IN
     APPENDIX III SHOW HOW TO IDENTIFY AND DISTINGUISH CUSPS AND POLES.

          -3-   WHICH FUNCTIONS ARE REPRODUCED EXACTLY BY THE SCHEME?    IN
PARTICULAR,  WHICH  IS  THE  MAXIMUM DEGREE UP TO WHICH ALL POLYNOMIALS ARE
REPRODUCED EXACTLY?  IN OTHER WORDS, WHAT IS THE  'DEGREE  OF  (POLYNOMIAL)
PRECISION' OF THE SCHEME?

          -4-   IF THE TRIAL FUNCTION (OR  SOME  OF  ITS  DERIVATIVES)  ARE
POLYNOMIAL WHAT IS THEIR DEGREE?

          IT SHOULD BE OBVIOUS THAT ONLY IN  THE  VERY  SIMPLEST  OF  CASES
ANSWERS  TO  THESE  QUESTIONS  CAN  BE OBTAINED FROM A PLAIN DISPLAY OF THE
SURFACE OF INTEREST.  EVEN GIVEN INFINITE  DISPLAY  ACCURACY  (ALTHOUGH  OF
COURSE  ANY  PHYSICAL  DISPLAY  IS  ULTIMATELY  PIECEWISE  LINEAR  OR  EVEN
DISCRETE),    IT    SEEMS   IMPOSSIBLE  TO  DISTINGUISH  OPTICALLY  A  ONCE
DIFFERENTIABLE FUNCTION FROM ONE THAT IS TWICE DIFFERENTIABLE.

                          1.2  THE BASIC APPROACH

          IN THE DESIGN OF AN APPROXIMATION SCHEME ONE  USUALLY  KNOWS  THE
'CRITICAL  SETS' WHERE THE PHENOMENA OF INTEREST SUCH AS DISCONTINUITIES IN
DERIVATIVES, OR INTERPOLATION TO CERTAIN DATA, MAY OCCUR.  FOR EXAMPLE,  IN
A  BIVARIATE INTERPOLATION SCHEME DEFINED ON A TRIANGULATION, CRITICAL SETS
ARE USUALLY EDGES OF TRIANGLES, POSSIBLY INTERNAL EDGES,  AND  VERTICES  OF
TRIANGLES.    IN  A  TRIVARIATE  SCHEME  DEFINED  ON  A  TESSELATION   INTO
TETRAHEDRA, CRITICAL SETS ARE FACES, EDGES, MAYBE INTERNAL FACES AND EDGES,

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AND  INTERNAL  AND  EXTERNAL VERTICES.  IF THE CRITICAL SET IS NOT KNOWN IT
CAN BE DISCOVERED WITH MICROSCOPE (SEE SECTION 3), BUT FOR THE  PRESENT  WE
WILL ASSUME IT IS KNOWN.

          THE BASIC IDEA OF MICROSCOPE IS VERY SIMPLE:  PICK A RANDOM POINT
(THE  'POINT OF EXAMINATION') IN A SPECIFIC CRITICAL SET.  WE WILL REFER TO
THAT SET AS 'THE FRONT'.  THEN DETERMINE A 'DIRECTION OF INVESTIGATION' AND
CONSIDER  A  LINE  THROUGH  THE  POINT  OF  EXAMINATION IN THE DIRECTION OF
INVESTIGATION.  THIS IS THE 'LINE OF INVESTIGATION'.   DEPENDING  UPON  THE
APPLICATION, THE LINE OF INVESTIGATION MAY BE CONTAINED IN THE FRONT, OR IT
MAY BE AT AN ANGLE (NOT NECESSARILY A RIGHT ONE) TO IT.  WE WILL  CALL  ANY
DERIVATIVE  IN  THE  DIRECTION  OF INVESTIGATION A 'TANGENTIAL DERIVATIVE'.
ANY    PURE   DERIVATIVE  IN  A  DIRECTION  OTHER  THAN  THE  DIRECTION  OF
INVESTIGATION  IS  A  'CROSS  DERIVATIVE'  AND  ITS DIRECTION IS THE 'CROSS
DIRECTION'.  MICROSCOPE NUMERICALLY APPROXIMATES THE  RELEVANT  DERIVATIVES
(PURE  OR  MIXED)  AND  DISPLAYS  THE  APPROXIMATIONS  ON THE CRT TERMINAL.
DEPENDING ON THE RESULTING PICTURE, ONE MAY THEN TAKE FURTHER ACTION,  E.G.
PICK  ANOTHER  POINT  OF EXAMINATION OR ANOTHER DIRECTION OF INVESTIGATION,
CONSIDER OTHER DERIVATIVES, OR IMPROVE THE EXISTING PLOT  BY  ALTERING  THE
UNDERLYING NUMERICAL PARAMETERS.

          IN USING MICROSCOPE IT IS  IMPORTANT  TO  CHOOSE  THE  UNDERLYING
DOMAIN  AND  THE  PRIMITIVE  FUNCTION  SO  AS TO AVOID ARTIFACTS DUE TO THE
REGULARITY OF THE DOMAIN OR PECULIARITIES OF THE PRIMITIVE FUNCTION, AND TO
PICK  THE POINTS OF EXAMINATION RANDOMLY OR ARBITRARILY WITHIN THE CRITICAL
SETS.  USUALLY DIFFERENT TYPES OF CRITICAL SETS  WILL  BE  PRESENT  AND  OF
COURSE ONE WILL NEED TO CONSIDER AT LEAST ONE REPRESENTATIVE OF EACH SET.

          NUMERICAL    DIFFERENTIATION  IS  A  NOTORIOUSLY  ILL-CONDITIONED
NUMERICAL  PROCESS.    HOWEVER,  IN MICROSCOPE, THE APPEARANCE OF ROUND-OFF
ERRORS IS IMMEDIATELY APPARENT BECAUSE THE  POINTS  ON  THE  GRAPH  OF  THE
RELEVANT  DERIVATIVE  ARE  SCATTERED ACROSS THE SCREEN IN A RANDOM FASHION.
MOREOVER,    BY  PLOTTING  DERIVATIVES  OF  A  HIGH  DEGREE  WITH  A  LARGE
DISCRETIZATION  PARAMETER  THE  LIMITATIONS IMPOSED BY ROUND-OFF ERRORS CAN
USUALLY BE OVERCOME.  FOR EXAMPLE,  IF  THE  EXPECTED  DISCONTINUITY  IN  A
SPECIFIC DERIVATIVE IS TOO UNPRONOUNCED TO BE VISIBLE AT THE LARGE VALUE OF
THE DISCRETIZATION PARAMETER NECESSITATED BY ROUND-OFF ERRORS, THEN  HIGHER
ORDER  DERIVATIVES  CAN  BE  COMPUTED  (ON AN EVEN COARSER DISCRETIZATION).
THESE   WILL  BE  APPROXIMATIONS  OF  THE  DIRAC  DELTA  FUNCTION  AND  ITS
DERIVATIVES, AND USUALLY WILL PINPOINT EVEN WEAK DISCONTINUITIES.

          NUMERICALLY, WE PROCEED AS FOLLOWS.    LET  F  DENOTE  THE  TRIAL
FUNCTION, P THE POINT OF EXAMINATION, AND D THE DIRECTION OF INVESTIGATION.
MICROSCOPE DISPLAYS DERIVATIVES OF F IN THE DIRECTION OF D ALONG  THE  LINE
OF INVESTIGATION WITH P BEING THE CENTER POINT.  A BUILT-IN FACILITY ALLOWS
THE REPLACEMENT OF F WITH A DERIVATIVE OF F  IN  SOME  OTHER,  INDEPENDENT,
DIRECTION.    MORE  COMPLICATED  MIXED  PARTIAL DERIVATIVES CAN BE BUILT BY
MODIFYING THE TRIAL FUNCTION, AND MICROSCOPE OFFERS SOME HELP WITH THAT  AS
WELL.    HOWEVER,  FOR  THE  PRESENT  WE  WILL  RESTRICT OUR ATTENTION TO A
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  11
TANGENTIAL DERIVATIVE OF F.

          THE DIRECTIONAL DERIVATIVES ARE APPROXIMATED AT POINTS

             P(I) = P + I*S*D, WHERE I = -N, -N+1, ...., N-1,N

AND THE QUANTITY S > 0 DETERMINES THE SPACING OF THE POINTS.   IT  WILL  BE
REFERED TO AS THE 'DISPLAY INTERVAL' OR 'INTERVAL' FOR SHORT.  THE NUMBER N
DETERMINES THE RESOLUTION OF THE DISPLAY (AS WELL AS THE NUMERICAL EFFORT).
AT  EACH  POINT  P(I),  WE  APPROXIMATE  THE  K-TH  DERIVATIVE  OF F IN THE
                                             (K)
DIRECTION D AT THE I-TH POINT BY A QUANTITY Q   .
                                             I

          THE  RANGE OF K IN THE PRESENT VERSION OF MICROSCOPE IS FROM ZERO
TO SIX.  WE HAVE FOUND THIS SUFFICIENT FOR ALL OF OUR  APPLICATIONS.    THE
             (K)
QUANTITIES  Q     ARE  DETERMINED  BY  NUMERICAL  DIFFERENTIATION (I.E.  BY
             I
DIFFERENTIATING AN INTERPOLATING POLYNOMIAL).  THE FORMULAS,  GIVEN  BELOW,
WERE DETERMINED BY THE FOLLOWING CRITERIA:

          -1-   CENTRAL DIFFERENCES SHOULD BE EMPLOYED IN  ORDER  TO  AVOID
ARTIFACT DUE TO ASYMMETRY.

          -2-   THE DISCRETIZATION PARAMETER (WHICH  DOES  NOT  NECESSARILY
EQUAL S) SHOULD BE SUCH THAT THE NUMBER OF POINTS AFFECTED BY THE SUSPECTED
DISCONTINUITY (IN SOME DERIVATIVE) IS  INDEPENDENT  OF  THE  ORDER  OF  THE
DERIVATIVE.

          -3-   OTHER THAN THE  ABOVE,  THE  DEGREE  OF  THE  INTERPOLATING
POLYNOMIAL SHOULD BE AS LOW AS POSSIBLE.

          FURTHERMORE, WE PICKED DISCRETIZATION FORMULAS THAT USE POINTS AT
EQUALLY  SPACED  INTERVALS.    THIS  MAY  BE  CHANGED IN FUTURE VERSIONS OF
MICROSCOPE.

          DENOTING  THE  'DISCRETIZATION  PARAMETER'  BY  H  > 0, THE ABOVE
CRITERIA GIVE RISE TO THE FOLLOWING FORMULAS:

           (0)
          Q    = F5
           I

           (1)
          Q    = (-F1 + F9)/(2*H)
           I

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  12
           (2)                    2
          Q    = (F1 - 2F5 + F9)/H
           I

           (3)                             3
          Q    = 4*(-F1 + 2F3 - 2F7 + F9)/H
           I

           (4)                                   4
          Q    = 16*(F1 - 4F3 + 6F5 - 4F7 + F9)/H
           I

           (5)                                             5
          Q    = 243*(-F1 + 4F2 - 5F4 + 5F6 - 4F8 + F9)/(2H )
           I

           (6)                                                    6
          Q    = 729*(F1 - 6F2 + 15F4 - -20F5 + 15F6 - 6F8 + F9)/H
           I

WHERE

          F1 = F(P(I) - HD)

                        2HD
          F2 = F(P(I) - ---)
                         3

                        HD
          F3 = F(P(I) - --)
                        2

                        HD
          F4 = F(P(I) - --)
                        3

          F5 = F(P(I))

                        HD
          F6 = F(P(I) + --)
                        3

                        HD
          F7 = F(P(I) + --)
                        2

                        2HD
          F8 = F(P(I) + ---)
                         3

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  13
          F9 = F(P(I) + HD)

          THE FIRST FOUR OF THE ABOVE FORMULAS CAN BE FOUND  IN  ABRAMOWITZ
AND  STEGUN,  1968,  P.    914,  AND  THE  LAST  TWO  CAN BE DERIVED BY THE
TECHNIQUES DESCRIBED THERE.  HIGHER ORDER FORMULAS CAN OF  COURSE  ALSO  BE
DERIVED.

                                          (K)
          IT IS INTERESTING TO NOTE THAT Q    CONSIDERED AS A FUNCTION OF X
                                          I
(OR  A FUNCTION OF A CONTINUOUSLY VARYING PARAMETER I) IS A FUNCTION OF THE
SAME DEGREE OF SMOOTHNESS AS THE TRIAL FUNCTION ITSELF.  JUST THE NUMBER OF
POINTS OF REDUCED SMOOTHNESS IS INCREASED, THERE BEING ONE CORRESPONDING TO
EACH POINT IN THE DIFFERENTIATION FORMULA  PASSING  THROUGH  THE  POINT  OF
REDUCED  SMOOTHNESS.  THUS NUMERICAL DIFFERENTIATION CAN BE THOUGHT OF AS A
SMOOTHING PROCESS.  (THIS CONTRASTS STARKLY WITH ITS NUMERICAL PROPERTY  OF
A  ROUGHING PROCEDURE DUE TO ROUND-OFF EFFECTS.) THE FIGURES IN SECTION 4.2
ILLUSTRATE THE SMOOTHING PROPERTIES OF NUMERICAL DIFFERENTIATION.

          FOR  A  MORE  DETAILED DISCUSSION OF THE TRUNCATION AND ROUND-OFF
PROPERTIES OF THESE FORMULAS SEE APPENDIX I.

                       1.3 THE ALPHANUMERICAL DISPLAY

          IN  THIS  SUBSECTION, THE ALPHANUMERICAL SCREEN DISPLAY GENERATED
BY MICROSCOPE IS DESCRIBED IN DETAIL.  THE PURPOSE OF  THE  DESCRIPTION  IS
TWOFOLD:    FIRST,  IT  IS  NECESSARY  FOR AN UNDERSTANDING OF THE EXAMPLES
SPRINKLED THROUGHOUT THIS MANUAL.  SECOND, HOWEVER, IT ALSO  SERVES  AS  AN
ILLUSTRATION  OF  SOME  OF  THE  CAPABILITIES  OF MICROSCOPE AND AS A FIRST
INTRODUCTION TO THE POWER AND VERSATILITY OF THE PACKAGE.

          DURING  A  MICROSCOPE  SESSION  THE  CRT  SCREEN  WILL USUALLY BE
OCCUPIED BY THE  'ALPHANUMERICAL  DISPLAY'  LIKE  THE  ONE  ILLUSTRATED  IN
FIGURE 1.  (MORE SOPHISTICATED GRAPHICAL DISPLAYS CAN ALSO BE OBTAINED, SEE
SECTIONS 4.14 AND 5.)

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  14
222                            I     :   **********************************
 . 22                          I     :  *  I                          22 .
  .  222                       I     : *   I                       222  .
   .    22                     I     :     I                     22    .
    .     22                   I     :     I                   22     .
     .      222                I     :*    I                222      .
      ..       22              I     :     I              22       ..
7654321..876543212287654321.987654321+123456789.123456782212345678..1234567
         .         222         I     :     I         222         .
          ..          22       I    *:     I       22          ..
            ..          22     I     :     I     22          ..
              ..          222  I     :     I  222          ..
                ...          22I   * :     I22          ...
                   ....        22 *  :    22        ....
**********************************22222222..........
===========================================================================
CD: DEG =  1 DIR = (  0.000000D+00,  1.000000D+00) CH  =   6.0000D-04
POINT            = (  0.000000D+00,  1.000000D+00)  S  =   5.0000D-03
DIRECTION        = (  1.000000D+00,  0.000000D+00)  H  =   3.0000D-02
F0 ( 9.39D-09, 1.27D-02) F2 ( 1.20D-01, 2.22D+00) F3 (-1.20D+01, 1.21D+01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     174

            FIGURE 1 DEMONSTRATION OF THE ALPHANUMERICAL DISPLAY

                                                              2
          IN  THIS  EXAMPLE,  THE TRIAL FUNCTION IS F(X,Y) = X *Y*ABS(X*Y).
WE ANALYZE A CROSS DERIVATIVE OF IT IN THE DIRECTION (0,1).  THE  DIRECTION
OF  INVESTIGATION IS (1,0), AND THE POINT OF EXAMINATION IS (0,1).  BECAUSE
OF THE ABSOLUTE VALUE TERM, THE CROSS DERIVATIVE IS  TWICE  BUT  NOT  THREE
TIMES TANGENTIALLY DIFFERENTIABLE AT THE POINT OF EXAMINATION.

          WE EXPLAIN THE DISPLAY STARTING AT THE TOP AND REFERRING TO LINES
EITHER BY THEIR POSITION OR BY THE FIRST WORD CONTAINED IN THEM.

          THE DISPLAY IS NATURALLY DIVIDED INTO TWO PARTS:   THE  FIRST  15
LINES  CONSTITUTE THE 'GRAPHICAL DISPLAY', AND LINES 17 THROUGH 21 FORM THE
'NUMERICAL DISPLAY'.  LINE 16 SEPARATES THE TWO  DISPLAYS  AND  IS  PRESENT
ONLY IF THERE IS SUFFICIENT SPACE AVAILABLE.

                        -1-   THE GRAPHICAL DISPLAY

          EVERY MICROSCOPE DISPLAY SHOWS THE GRAPHS OF A SET OF  TANGENTIAL
DERIVATIVES  OF  DEGREES  BETWEEN  0  AND  6.    THE FUNCTION THAT IS BEING
DIFFERENTIATED TANGENTIALLY IS CALLED THE 'DISPLAY FUNCTION'.   IT  MAY  BE
THE  TRIAL  FUNCTION  ITSELF (THE MOST FREQUENTLY OCCURING CASE), OR, AS IN
THIS EXAMPLE, A PURE CROSS DERIVATIVE OF A DEGREE BETWEEN 1 AND 6.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  15
          DISTANCE ALONG THE HORIZONTAL AXIS CORRESPONDS TO DISPLACEMENT IN
THE DIRECTION  OF  INVESTIGATION  FROM  THE  POINT  OF  EXAMINATION  (WHICH
CORRESPONDS  TO THE CENTER OF THE HORIZONTAL AXIS).  THE SCALE ON THAT AXIS
HAS THE SAME MEANING FOR ALL GRAPHS SHOWN.  THE VERTICAL  AXIS  CORRESPONDS
TO  FUNCTION  VALUES,  BUT  THE  VERTICAL  SCALE  IS  DIFFERENT  FOR   EACH
DERIVATIVE, AND IS CHOSEN SUCH THAT THE GRAPH  FILLS  THE  ENTIRE  VERTICAL
EXTENT  OF  THE  GRAPHICAL  DISPLAY.  THIS NORMALIZATION IS CARRIED OUT AND
MODIFIED AUTOMATICALLY AS THE INVESTIGATION PROCEEDS.  A PERIOD DENOTES THE
GRAPH  OF  THE DISPLAY FUNCTION, AND THE DIGIT K, SAY, WHERE K IS BETWEEN 1
AND 6 AND DENOTES THE K-TH TANGENTIAL DERIVATIVE.   IN  THIS  EXAMPLE,  THE
DISPLAY  FUNCTION AND ITS SECOND AND THIRD TANGENTIAL DERIVATIVES ARE BEING
SHOWN.

          SOMETIMES  SOME  DERIVATIVE  IS MORE INTERESTING THAN OTHERS, AND
THEN IT MAY BE USEFUL TO ACCENTUATE  IT  WITH  ASTERISKS.    IN  THE  ABOVE
EXAMPLE,  THE  THIRD DERIVATIVE HAS BEEN SO EMPHASIZED.  USUALLY, THE GRAPH
OF A GIVEN DERIVATIVE OVERWRITES THE GRAPH OF ANY LOWER  ORDER  DERIVATIVE.
HOWEVER,  THE  GRAPH  OF  AN  ACCENTUATED  DERIVATIVE  OVERWRITES ALL OTHER
GRAPHS.

          BOTH THE NUMBER OF LINES AND THE NUMBER OF COLUMNS IN THE DISPLAY
CAN BE SET ACCORDING TO SPECIFIC PURPOSES.  THROUGHOUT THIS MANUAL WE  WILL
USE 75 COLUMNS AND 15 ROWS FOR THE GRAPHICAL DISPLAY AS THIS IS APPROPRIATE
FOR THE WIDTH OF THE PRINTED OUTPUT.  A WIDTH OF 79 MIGHT BE PREFERABLE FOR
A  STANDARD  CRT  TERMINAL  DISPLAYING 80 COLUMNS; 135 COLUMNS AND A LARGER
NUMBER OF ROWS MIGHT BE USED FOR PRINTED OUTPUT; AND A  SMALLER  NUMBER  IS
SOMETIMES  USEFUL FOR PRELIMINARY INVESTIGATIONS WHEN THE TRIAL FUNCTION IS
EXPENSIVE TO EVALUATE.  AN EVEN NUMBER OF COLUMNS IS  NORMALLY  UNDESIRABLE
BECAUSE THEN THE POINT OF EXAMINATION DOES NOT LIE IN THE PRECISE CENTER OF
THE DISPLAY.

          LINE  8  IN THE EXAMPLE CONTAINS A HORIZONTAL SCALE COUNTING FROM
THE CENTER TO THE LEFT AND RIGHT.  THIS WILL USUALLY BE OMITTED SO  AS  NOT
TO  CLUTTER  THE DISPLAY, BUT IT IS SOMETIMES USEFUL FOR THE IDENTIFICATION
OF POINTS OF INTEREST.  THE DISTANCE FROM THE CENTER CAN BE USED TO INQUIRE
ABOUT  NUMERICAL  VALUES OF CERTAIN DERIVATIVES, OR TO SHIFT THE GRAPH LEFT
OR RIGHT AS REQUIRED BY THE CIRCUMSTANCES.  THE SCALE CAN BE TURNED  ON  OR
OFF,  OR  REPLACED  WITH  A  LINE  OF  HYPHENS.   THE GRAPH OF ANY FUNCTION
OVERWRITES THE HORIZONTAL SCALE.  THE CENTER OF THE DISPLAY CAN  OPTIONALLY
BE  MARKED  WITH A "+" SIGN.  THIS IS SOMETIMES USEFUL TO PINPOINT SYMMETRY
PROPERTIES.  IN THE ABOVE EXAMPLE THE GRAPH OF THE THIRD DERIVATIVE  PASSES
THROUGH THE CENTER OF THE DISPLAY.  THE CENTER MARK OVERWRITES THE GRAPH OF
ANY FUNCTION PASSING THROUGH IT.

          THE  CENTRAL  COLUMN  OF COLONS MARKS THE COLUMN CORRESPONDING TO
THE POINT OF EXAMINATION.  THE TWO COLUMNS OF  I'S  OUTLINE  'THE  WINDOW',
I.E.  THAT SET OF POINTS AT WHICH THE NUMERICAL DIFFERENTIATION IS AFFECTED
BY THE BEHAVIOR OF THE TRIAL FUNCTION AT THE POINT OF EXAMINATION.    (THIS
IS NOT ALWAYS QUITE TRUE WHEN CROSS DERIVATIVES ARE BEING EXAMINED, SEE THE
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  16
DISCUSSION IN SECTION 4.11).  FOR EVERY POINT IN THE WINDOW,  THE  INTERVAL
ABOUT  THE  POINT SPANNED BY THE DIFFERENTIATION STENCIL CONTAINS THE POINT
OF EXAMINATION.  THE DIFFERENTIATION FORMULAS  ARE  CHOSEN  SUCH  THAT  THE
WINDOW  IS  IDENTICAL  FOR  ALL TANGENTIAL DERIVATIVES.  OBVIOUSLY, FOR THE
DISPLAY FUNCTION ITSELF THE WINDOW CONTAINS ONLY THE POINT OF  EXAMINATION.
FOR  ITS DERIVATIVES, THE WINDOW CONTAINS ALL POINTS THAT CAN BE WRITTEN AS
P + TD, WHERE ABS(T) DOES NOT EXCEED THE VALUE OF H/S, (AND, AS  BEFORE,  P
IS  THE POINT OF EXAMINATION, D IS THE DIRECTION OF INVESTIGATION, H IS THE
DISCRETIZATION PARAMETER, AND S IS THE INTERVAL).  WE DEFINE W:=2*H/S TO BE
THE  'WINDOW  WIDTH'.    OBVIOUSLY,  BY PICKING THE WINDOW WIDTH < 1, IT IS
POSSIBLE TO GENERATE DISPLAYS IN WHICH THE  ONLY  POINT  CONTAINED  IN  THE
WINDOW  IS P ITSELF.  HOWEVER, FOR SEVERAL REASONS IT IS USUALLY PREFERABLE
TO USE A LARGER WINDOW WIDTH.  THIS ASPECT IS DISCUSSED IN MORE  DETAIL  IN
SECTION  4.1.    THE  DEFAULT  VALUE  OF  THE WINDOW WIDTH IS 12, AS IN THE
EXAMPLE, BUT THIS CAN BE CHANGED AS NEEDED.

          WITHOUT  EXAMINING THE NUMERICAL DISPLAY, IT IS APPARENT FROM THE
GRAPHICAL DISPLAY THAT SOME FUNCTION IS BEING INVESTIGATED  THAT  IS  TWICE
BUT NOT THREE TIMES DIFFERENTIABLE SINCE THE (ACCENTUATED) THIRD DERIVATIVE
SHOWS A CLEAR STEP DISCONTINUITY.  NOTE HOW THAT STEP IS SMEARED  OUT  OVER
THE WINDOW.

                         -2-  THE NUMERICAL DISPLAY

          THE FIRST LINE OF  THE  NUMERICAL  DISPLAY  STARTING  WITH  "CD:"
INDICATES  THAT  THE  DERIVATIVES  ARE  MIXED  PARTIALS,  I.E.    THEY  ARE
TANGENTIAL DERIVATIVES OF A CROSS DERIVATIVE (OF THE TRIAL FUNCTION).  THIS
LINE IS ABSENT IF THE TRIAL FUNCTION ITSELF IS BEING EXAMINED, AND THE LINE
CARRIES ONLY INFORMATION THAT IS PERTINENT TO THE CROSS DERIVATIVE.  IN THE
PRESENT EXAMPLE, THE DEGREE OF THE CROSS DERIVATIVE ("DEG") IS 1, THE CROSS
DIRECTION IS (0,1), AND THE DISCRETIZATION PARAMETER USED FOR COMPUTING THE
DERIVATIVE  (DENOTED  BY  "CH"  CORRESPONDING  TO  "H"  FOR  THE TANGENTIAL
DERIVATIVE) IS 6D-4.  THE FORMULAS  FOR  COMPUTING  CROSS  DERIVATIVES  ARE
EQUIVALENT TO THOSE FOR COMPUTING TANGENTIAL DERIVATIVES.

          THE LINE STARTING WITH "POINT" DESCRIBES THE POINT OF EXAMINATION
(WHICH  IS  (0,1))  AND THE INTERVAL (WHICH IS 5D-3).  THIS DISPLAY IMPLIES
THAT THE TRIAL FUNCTION IS BIVARIATE.  APPROPRIATE MODIFICATIONS TAKE PLACE
FOR FUNCTIONS OF 1 OR 3 VARIABLES.  THE NEXT LINE STARTING WITH "DIRECTION"
DESCRIBES   THE  DIRECTION  OF  INVESTIGATION  (WHICH  IS  (1,0))  AND  THE
DISCRETIZATION  PARAMETER (WHICH IS 3D-2).  NOTICE THAT THE WINDOW WIDTH IS
2H/S = 12 WHICH IS CONSISTENT WITH THE GRAPHICAL DISPLAY

          THE "POINT" AND "DIRECTION" DESCRIPTIONS ARE ALWAYS PRESENT.

          THE NEXT LINE, STARTING  WITH  "F0",  GIVES  THE  RANGES  OF  THE
DISPLAYED  DERIVATIVES.   RANGES ARE LABELED FK WHERE K IS THE ORDER OF THE
TANGENTIAL DERIVATIVE.  FOR EXAMPLE, F2 CORRESPONDS TO  THE  RANGE  OF  THE
SECOND  (TANGENTIAL)  DERIVATIVE  (OF A FIRST ORDER CROSS DERIVATIVE) WHICH

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  17
HERE IS (0.12,2.1).  IF MORE FUNCTIONS (UP TO SEVEN)  ARE  DISPLAYED,  THEN
THEIR  RANGES  ARE  ALSO  GIVEN,  ADDING  UP  TO TWO LINES TO THE NUMERICAL
DISPLAY AS NECESSARY.

          THE NEXT AND LAST LINE, STARTING WITH "I/O", CONTAINS INFORMATION
ABOUT THE PRESENT CONFIGURATION OF MICROSCOPE.

          THE  10  INTEGERS FOLLOWING "I/O" ARE THE FORTRAN CHANNEL NUMBERS
AS THEY ARE EMPLOYED BY MICROSCOPE AT THE DISPLAY TIME.  THEY CORRESPOND TO
THE FOLLOWING:

          -1-   'INPUT DEVICE' (HERE 25),  I.E.    THE  DEVICE  FROM  WHICH
COMMANDS  ARE  READ.    IN AN INTERACTIVE SESSION, THIS WILL USUALLY BE THE
TERMINAL.  BUT IT MAY ALSO BE A FILE OF MICROSCOPE COMMANDS THAT GENERATE A
SPECIFIC SETUP, OR A DOCUMENTATION LIKE THIS MANUAL.

          -2-   'OUTPUT DEVICE' (HERE 6), I.E.  THE  DEVICE  RECEIVING  THE
COMMAND  PROMPTS  AND ANY HELP INFORMATION.  USUALLY, THIS WILL ALSO BE THE
TERMINAL, OR, IF MICROSCOPE IS BEING DRIVEN BY COMMANDS IN A FILE,  IT  MAY
BE A NULL OR DUMMY DEVICE.

          -3-   'GRAPHICS DEVICE' (HERE ALSO 6), I.E.  THE DEVICE RECEIVING
THE  ALPHANUMERICAL  DISPLAY  ILLUSTRATED  IN  FIGURE  1.   THIS WILL AGAIN
USUALLY BE THE TERMINAL, BUT IT MAY ALSO BE A SEPARATE DISPLAY DEVICE OR  A
FILE RECEIVING INFORMATION FOR SUBSEQUENT PRINTING.

          -4-   'HELP DEVICE' (HERE 1), I.E.   THE  DEVICE  CONTAINING  THE
MICROSCOPE  HELP  FILE, WHICH IS SUPPLIED WITH THE PACKAGE.  MICROSCOPE HAS
SEVERAL INTERACTIVE HELP FACILITIES AND THE INFORMATION NEEDED FOR  RUNNING
THEM IS READ OUT OF THIS FILE WHEN MICROSCOPE IS FIRST CALLED.  USUALLY YOU
WILL NOT BE CONCERNED WITH PRECISELY WHERE  MICROSCOPE  RECEIVES  ITS  HELP
INFORMATION.  HOWEVER, THE CHANNEL NUMBER IS PRINTED IN THE DISPLAY TO HELP
PREVENT AN ACCIDENTAL OVERWRITE OF THE HELP FILE.

          -5-   'RECORDING  DEVICE'  (HERE  2),  A  DEVICE THAT CAN RECEIVE
SELECTED INFORMATION FROM THE MICROSCOPE  SESSION,  E.G.    COPIES  OF  THE
SCREEN  DISPLAY  OR  'NOTES' THAT EXPLAIN THE DISPLAYS.  THIS MANUAL IS THE
RESULT OF READING A SET OF INSTRUCTIONS, I.E.  A MICROSCOPE 'PROGRAM'  FROM
THE  INPUT DEVICE (A DATA FILE), PROCESSING IT AND WRITING THE RESULTS ONTO
THE RECORDING DEVICE.

          -6-   'RESTART  DEVICE'  (HERE 3).  AT ANY TIME DURING A SESSION,
THE CURRENT CONFIGURATION OF MICROSCOPE CAN  BE  SAVED,  ALLOWING  A  LATER
RESTART FROM THE CURRENT STATUS OF THE INVESTIGATION.  THIS IS ACCOMPLISHED
BY WRITING THE VALUES OF ALL RELEVANT VARIABLES INTO A FILE POINTED  AT  BY
THE RESTART DEVICE.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  18
          -7-   'LOGGING DEVICE' (HERE 27).  IT IS POSSIBLE TO KEEP A DIARY
OF  TH  MICROSCOPE SESSION LISTING THE COMMANDS AND DATA ENTERED AS WELL AS
TH RESULTS EFFECTED BY THEM.  THIS FEATURE  IS  ACTIVATED  BY  SETTING  THE
LOGGING DEVICE NUMBER TO A NON-ZERO VALUE.

          -8-   'POINT LOADING DEVICE' (HERE 28).  A DEVICE ALLOWING YOU TO
READ POINTS FROM A FILE, RATHER THAN HAVING TO TYPE THEM IN.

          -9-   'DIRECTION LOADING DEVICE' (HERE 29).  SIMILAR TO  -8-  FOR
LOADING THE DIRECTION OF INVESTIGATION.

          -10-  'CROSS DIRECTION LOADING DEVICE' (HERE 30).  SIMILAR TO -8-
FOR LOADING THE CROSS DIRECTION.

          THE FIRST SIX I/O CHANNEL NUMBERS ARE PARAMETERS  OF  THE  MASTER
MICROSCOPE  ROUTINE  (SEE  SECTION  3.1)  AND  CAN EACH BE CHANGED DURING A
SESSION.  THE LAST FOUR CHANNELS ARE ZERO (I.E.  UNUSED)  BY  DEFAULT,  AND
CAN  BE  SET TO NON-ZERO (I.E.  ACTIVE) VALUES DURING A MICROSCOPE SESSION.
IT IS OF COURSE UNNECESSARY TO HAVE ALL  CHANNEL  NUMBERS  DISTINCT.    FOR
EXAMPLE,  ALL  THREE  LOADING  DEVICES MIGHT BE IDENTICAL, ALLOWING FOR THE
POSSIBILITY OF HAVING JUST  ONE  LIST  OF  POINTS,  DIRECTIONS,  AND  CROSS
DIRECTIONS.

          FOLLOWING THE DEVICE NUMBERS IS A FLAG "NRML ON"  OR  "NRML  OFF"
(HERE  ON),  INDICATING WHETHER THE DIRECTION OF INVESTIGATION ENTERED INTO
MICROSCOPE HAS BEEN NORMALIZED OR NOT.  THE "DIRECTION" LINE  ALWAYS  GIVES
THE  DIRECTION  AS  ENTERED  BY THE USER.  IF NRML IS OFF, THE DIRECTION OF
INVESTIGATION, D, IS IDENTICAL TO THAT PRINTED IN THE DISPLAY, OTHERWISE IT
IS  THAT  PRINTED  IN THE DISPLAY NORMALIZED TO HAVE UNIT EUCLIDEAN LENGTH.
WITH   THE  NORMALIZATION  ON,  TANGENTIAL  DERIVATIVES  ARE  THE  STANDARD
DIRECTIONAL  DERIVATIVES  AS  THEY ARE DEFINED IN CALCULUS TEXTBOOKS.  WITH
THE NORMALIZATION OFF, THEY ARE THE MORE VERSATILE GATEAUX  DERIVATIVES  AS
THEY  ARE  USED FREQUENTLY IN MULTIVARIATE INTERPOLATION AND APPROXIMATION.
THE NRML FLAG APPLIES SIMILARLY TO THE CROSS DIRECTION.

          FOLLOWING  THE  NRML  FLAG IS THE WORD "CURRENT".  THIS INDICATES
THAT THE NUMERICAL DISPLAY CORRESPONDS TO THE GRAPHICS DISPLAY.  TO EXPLAIN
WHY  THIS  MIGHT NOT BE SO REQUIRES A BRIEF EXCURSION INTO THE ORGANIZATION
OF MICROSCOPE.  WHEN DRASTIC PARAMETER CHANGES ARE TAKING PLACE IT IS OFTEN
MORE  EFFICIENT  TO  ACCUMULATE  THEM  BEFORE  GENERATING  THE NEW DISPLAY.
USUALLY IN THAT CASE THE ALPHANUMERICAL DISPLAY IS SCROLLED WHILE  COMMANDS
ARE  BEING  GIVEN,  AND VANISHES FROM THE SCREEN.  HOWEVER, SOMETIMES IT IS
PREFERABLE TO VIEW THE OLD DISPLAY RATHER THAN THE COMMANDS GIVEN  SO  FAR.
IN  THAT  CASE,  THE  DISPLAY  CAN  BE RECALLED ON THE SCREEN, AND THE WORD
"CURRENT" IS REPLACED BY "GO PNDG" (I.E.  THE GO COMMAND NEEDS TO BE  GIVEN
TO  INCORPORATE  INTO  THE GRAPHICAL DISPLAY CHANGES OF PARAMETERS THAT MAY
ALREADY BE SHOWN IN THE NUMERICAL DISPLAY).

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  19
          THE  LAST ITEM IN THE LAST LINE IS "CALLS = 174" WHERE 174 IS THE
NUMBER OF EVALUATIONS OF THE TRIAL FUNCTION EXPENDED UP TO  THIS  POINT  IN
THE  MICROSCOPE  SESSION.   THIS ALLOWS MONITORING THE COMPUTATIONAL EFFORT
SPENT.  A TYPICAL MULTIVARIATE SURFACE WILL BE EXPENSIVE TO  EVALUATE,  AND
ITS  COST  WILL USUALLY DOMINATE THE COST OF RUNNING MICROSCOPE AND THEREBY
DETERMINE HOW LONG YOU HAVE TO  WAIT  FOR  A  REPONSE  FROM  THE  TERMINAL.
THEREFORE,  A  SIGNIFICANT  EFFORT  HAS  BEEN  MADE  TO  KEEP THE NUMBER OF
NECESSARY FUNCTION EVALUATIONS SMALL.  THIS FEATURE CONTRIBUTES IN NO SMALL
MEASURE  TO  THE  COMPLEXITY  OF  THE  MICROSCOPE SOURCE CODE.  THE EXAMPLE
DISPLAY HAS 75 COLUMNS AND WE ARE EXAMINING A FIRST ORDER CROSS  DERIVATIVE
OF  THE  TRIAL  FUNCTION.  EACH EVALUATION OF THE CROSS DERIVATIVE REQUIRES
TWO EVALUATIONS OF THE TRIAL FUNCTION.  DISPLAYING THE GRAPH OF  THE  CROSS
DERIVATIVE  ALONE  WOULD  THEREFORE  REQUIRE  A  VALUE  OF  CALLS  EQUAL TO
2*75 = 150.  NOTE THAT  COMPUTING  AND  DISPLAYING  THE  SECOND  AND  THIRD
TANGENTIAL  DERIVATIVES  OF  THE  CROSS  DERIVATIVE  REQUIRES AN ADDITIONAL
EFFORT OF ONLY 16%.  THIS  IS  NOT  A  TRIVIAL  ACCOMPLISHMENT  AS  MAY  BE
SUGGESTED  BY  THE  FACT  THAT THE SAME DISPLAY WITH THE SLIGHTLY DIFFERENT
WINDOW WIDTH  OF  10  WOULD  REQUIRE  330  EVALUATIONS.    FOR  A  DETAILED
DISCUSSION  OF  THE  NUMERICAL EFFORT ASSOCIATED WITH VARIOUS WINDOW WIDTHS
SEE SECTION 4.3.

          NOTICE  THAT  IN  THE  ABOVE  EXAMPLE  WE  IDENTIFIED   A    STEP
DISCONTINUITY IN A MIXED PARTIAL DERIVATIVE OF FOURTH ORDER (A THIRD  ORDER
TANGENTIAL  DERIVATIVE  OF  A FIRST ORDER CROSS DERIVATIVE) WITHOUT FUSS OR
DOUBT.  IF THIS WAS A REAL  INVESTIGATION  WE  COULD  STATE  NOW  THAT  THE
UNDERLYING SCHEME IN ANY CASE IS AT MOST THREE TIMES DIFFERENTIABLE.

                                1.4 SUMMARY

          WITH MICROSCOPE, ONE ANSWERS QUESTIONS LIKE THOSE IN SECTION  1.1
FOR  SPECIFIC EXAMPLES.  THUS ONLY NEGATIVE RESULTS (E.G.  THAT A SCHEME IS
NOT DIFFERENTIABLE) CAN BE PROVED STRICTLY.  HOWEVER, POSITIVE RESULTS  CAN
ALSO  BE  OBTAINED.    IF  THE  EXAMPLES  ARE  CHOSEN  CAREFULLY TO EXCLUDE
ARTIFACTS AND MAKE THEM REPRESENTATIVE, OUR PROGRAM ALLOWS FOR  ANSWERS  TO
THE  ABOVE QUESTIONS WITH A DEGREE OF CONFIDENCE THAT BORDERS ON CERTAINTY!
NATURALLY, A COMPUTATIONAL APPROACH DOES  NOT  REPLACE  A  RIGOROUS  PROOF.
HOWEVER,  IN  THE  AUTHORS  EXPERIENCE  IT IS OFTEN POSSIBLE TO CONFIRM (OR
SHATTER) FUZZY NOTIONS WITH MICROSCOPE, AND  TO  GENERATE  FURTHER  INSIGHT
INTO THE PHENOMENA UNDER CONSIDERATION.  ENCOURAGING MICROSCOPE RESULTS MAY
HELP IN BUILDING UP THE STAMINA NEEDED FOR ATTEMPTING A RIGOROUS PROOF, AND
DETAILED  INVESTIGATIONS  MAY  HELP IN PROVIDING IDEAS FOR CARRYING OUT THE
ANALYSIS.

          OFTEN,  OF COURSE, MICROSCOPE WILL BE EMPLOYED IN HINDSIGHT, WHEN
THE ANALYST ALREADY KNOWS THE ANSWERS TO  QUESTIONS  LIKE  THE  ABOVE  FROM
THEORETICAL  REASONING.    THE  EXAMINATION  WITH MICROSCOPE THEN SERVES TO
CORROBORATE    THE   REASONING,  AND  ALSO,  MORE  MUNDANELY  BUT  NO  LESS
IMPORTANTLY, TO ELIMINATE BUGS IN THE IMPLEMENTATION.
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  20

          IN THE AUTHORS EXPERIENCE,  MICROSCOPE  PROVED  INVALUABLE  AS  A
DEBUGGING  TOOL, OFTEN PINPOINTING WITHIN NARROW LIMITS THAT PART OF A CODE
THAT WAS AT FAULT, BUT ALSO SOMETIMES UNCOVERING FLAWS IN  THE  THEORETICAL
APPROACH, AND, ON A FEW EXHILARATING OCCASIONS, LEADING TO THE DISCOVERY OF
NEW AND UNLOOKED FOR RESULTS THAT COULD THEN BE PROVED THEORETICALLY.  THUS
MICROSCOPE  IS  USEFUL IN THE IDENTIFICATION AS WELL AS IN THE VERIFICATION
OF RESULTS.

          IN  THE DEVELOPMENT OF MICROSCOPE WE WERE OCCASIONALLY CONFRONTED
WITH THE OBJECTION THAT OUR ALPHANUMERICAL DISPLAYS ARE RATHER CRUDE.   OUR
PRIMARY RESPONSE TO THIS IS THAT THE VERY PURPOSE OF MICROSCOPE CONSISTS OF
TRANSLATING VERY SUBTLE PROPERTIES INTO DISPLAYS THAT MAKE  THEM  GLARINGLY
APPARENT.   A LARGE PART OF OUR INTEREST CENTERS AROUND STEP FUNCTIONS, AND
IT SEEMS FAIR TO DISPLAY THEM AS SUCH.  INDEED,  ANY  CURVE  FITTING  MIGHT
DISGUISE  THE  VERY  FEATURES  IN  WHICH WE ARE INTERESTED.  HOWEVER, WE DO
PROVIDE AN INTERFACE TO A SOPHISTICATED GRAPHICS  PACKAGE.    SEE  SECTIONS
4.14 AND 5 FOR MORE INFORMATION.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  21
                              2. APPLICATIONS

          IN    THIS  SECTION,  WE  ILLUSTRATE  THE  MAIN  APPLICATIONS  OF
MICROSCOPE.    CONSIDER  THE SIMPLE EXAMPLE OF INTERPOLATING TO A PRIMITIVE
FUNCTION P BY A CUBIC SPLINE S, SAY, ACCORDING TO THE CONDITIONS:

                          2       3
   S(X) = A0 + A1*X + A2*X  + A3*X       IF X < 1
AND
                          2       3
   S(X) = B0 + B1*X + B2*X  + B3*X       OTHERWISE

WHERE THE COEFFICIENTS ARE DEFINED BY

       S(0) = P(0), S(1) = P(1), S(2) = P(2), S'(2) = P'(2), S"(0) = 0

AND S IS TWICE DIFFERENTIABLE AT X = 1.  THUS WE REQUIRE  INTERPOLATION  TO
'POSITION' (I.E.  FUNCTION VALUES) AT 0,1,2, AND INTERPOLATION TO THE FIRST
DERIVATIVE AT 2 BY A PIECEWISE CUBIC FUNCTION THAT  IS  TWICE  CONTINUOUSLY
DIFFERENTIABLE.    AT  0  WE  IMPOSE  ADDITIONALLY  A "FREE END" CONDITION.
(THERE IS AN EXTENSIVE LITERATURE ON SPLINES, SEE E.G.  DE BOOR,  1978,  OR
SCHUMAKER, 1981)

          FOR OUR PRESENT PURPOSES WE ASSUME THAT P(X) IS THE  EXPONENTIAL,
I.E.

                                         X
                                 P(X) = E

WHERE E IS THE BASE OF THE NATURAL LOGARITHM.  THIS FUNCTION APPEARS TO  BE
SUFFICIENTLY NON-POLYNOMIAL TO AVOID THE INTRODUCTION OF ARTIFACTS.

          A SIMPLE CALCULATION SHOWS THAT

          A0 = 1
                    2
          A1 = (-2*E +12*E-9)/7

          A2 = 0
                   2
          A3 = (2*E -5*E+2)/7

          AND
                   2
          B0 = (5*E -16*E+12)/7
                     2
          B1 = (-17*E +60*E-24)/7

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  22
                    2
          B2 = (15*E -48*E+15)/7
                    2
          B3 = (-3*E +11*E-3)/7

          IN THE FOLLOWING EXAMPLES WE WILL VERIFY THAT S  WITH  THE  ABOVE
COEFFICIENTS DOES INDEED POSSESS THE REQUIRED PROPERTIES.

                             2.1  INTERPOLATION

          THE  BASIC  IDEA  OF  VERIFYING  INTERPOLATION  IS  TO  PLOT  THE
DIFFERENCE BETWEEN THE PRIMITIVE FUNCTION AND ITS INTERPOLANT, AND TO CHECK
THAT THE ERROR AND ITS APPROPRIATE DERIVATIVES ARE  INDEED  ZERO.    SO  WE
INCORPORATE  THE  EXPONENTIAL  AND  S AS DEFINED ABOVE INTO MICROSCOPE (SEE
SECTION 3.1 FOR DETAILS), AND START AN INTERACTIVE SESSION.   AFTER  GIVING
THE  APPROPRIATE  COMMANDS WE OBTAIN THE FOLLOWING PICTURE ILLUSTRATING THE
SITUATION AT X = 0:

...                            I     :     I
   ...                         I     :     I
      ....                     I     :     I
          ...                  I     :     I
             ...               I     :     I
                ....           I     :     I
                    ....       I     :     I
                        ....   I     +     I
                            .....    :     I
                               I .....     I
                               I     :.....I
                               I     :     ......
                               I     :     I     .......
                               I     :     I            .........
                               I     :     I                     ..........
===========================================================================
POINT            =    0.000000D+00  S  =   5.0000D-03
DIRECTION        =    1.000000D+00  H  =   3.0000D-02
F0 (-3.33D-02, 6.76D-02)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     249

                 FIGURE 2 INTERPOLATION TO POSITION AT ZERO

          WE  NOTICE THAT THE RANGE OF THE ERROR IS FROM -0.0333 TO 0.0676.
SINCE THE GRAPH OF THE ERROR FUNCTION  PASSES  THROUGH  THE  VERTICAL  AXIS
SLIGHTLY  BELOW  THE  CENTER, THIS RANGE IS CONSISTENT WITH THE ERROR BEING
ZERO, I.E.  WITH INTERPOLATION.    A  MORE  RELIABLE  CONFIRMATION  CAN  BE
OBTAINED  BY  ASKING  FOR THE VALUE OF THE DISPLAYED FUNCTION AT THE CENTER
(I.E.  0) OF THE DISPLAY.  THIS PROCEDURE, WHICH IS NOT SHOWN HERE,  YIELDS
A  VALUE OF -9.4D-11, WHICH IS ZERO UP TO ROUND-OFF ERRORS.  (IF YOU TRY TO
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  23
REPRODUCE THIS EXAMPLE WITH YOUR OWN VERSION OF MICROSCOPE YOU  MAY  GET  A
SLIGHTLY  DIFFERENT  RESULT  BECAUSE  OF  THE  RANDOM  NATURE  OF ROUND-OFF
ERRORS.) AN ALTERNATIVE APPROACH CONSISTS  OF  HALVING  THE  INTERVAL,  AND
THEREBY  DECREASING  THE  RANGE  COVERED  BY  THE  GRAPH.  IN THE NUMERICAL
DISPLAY THE RANGE SHOULD SHRINK CORRESPONDINGLY, MAINTAINING THE  VALUE  OF
ZERO IN ITS INTERIOR.  HALVING THE INTERVAL YIELDS THE DISPLAY:

....                           I     :     I
    ....                       I     :     I
        ....                   I     :     I
            ....               I     :     I
                ....           I     :     I
                    .....      I     :     I
                         ....  I     :     I
                             .....   +     I
                               I  .....    I
                               I     : .....
                               I     :     I.....
                               I     :     I     ......
                               I     :     I           ......
                               I     :     I                 .......
                               I     :     I                        .......
===========================================================================
POINT            =    0.000000D+00  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 (-2.03D-02, 2.88D-02)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     287

                                  FIGURE 3

          THE RANGE OF THE ERROR HAS BEEN REDUCED TO FROM -0.0203 TO 0.0288
AND  A  ZERO  VALUE  OF  THE  ERROR  AT  THE  POINT OF EXAMINATION IS STILL
CONSISTENT WITH THE GRAPHICAL DISPLAY.  THIS PROCEDURE  CAN  BE  CONTINUED.
PROCEEDING  SIMILARLY  AT  THE POINT X = 1 YIELDS A SIMILAR RESULT, AND YOU
MIGHT TRY THIS FOR YOURSELF.  AT THE POINT X = 2, HOWEVER,  WE  OBTAIN  THE
DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  24
                               I     :     I                             ..
                               I     :     I                            .
                               I     :     I                           .
                               I     :     I                          .
                               I     :     I                        ..
                               I     :     I                       .
                               I     :     I                     ..
..                             I     +     I                    .
  ...                          I     :     I                  ..
     ....                      I     :     I                ..
         ...                   I     :     I              ..
            ....               I     :     I           ...
                .....          I     :     I        ...
                     .....     I     :     I    ....
                          ......................
===========================================================================
POINT            =    2.000000D+00  S  =   5.0000D-03
DIRECTION        =    1.000000D+00  H  =   3.0000D-02
F0 ( 2.64D-11, 1.42D-02)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     362

                      FIGURE 4 INTERPOLATION AT X = 2

          THE GRAPH OF THE ERROR FUNCTION TOUCHES, BUT DOES NOT CROSS,  THE
X  AXIS, WHICH IS CONSISTENT WITH OUR INTERPOLATING TO THE FIRST DERIVATIVE
AS WELL AS POSITION.  THE VALUE OF THE ERROR AT  THE  CENTER  CAN  BE  READ
DIRECTLY FROM THE NUMERICAL DISPLAY AND IS -2.4D-11 (YOU MAY GET A SLIGHTLY
DIFFERENT VALUE BECAUSE OF THE RANDOM ROUND-OFF  ERRORS).    THIS  CONFIRMS
INTERPOLATION  TO  POSITION.    INTERPOLATION  TO  THE DERIVATIVE IS LIKELY
BECAUSE  THE  GRAPH  APPEARS  TO  HAVE  A  HORIZONTAL  TANGENT.    HOWEVER,
CONFIDENCE  CAN BE RAISED BY COMPUTING THE FIRST DERIVATIVE OF THE ERROR AS
WELL, YIELDING THE DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  25
                               I     :     I                            111
                               I     :     I                          11.
                               I     :     I                       111 .
                               I     :     I                    111   .
                               I     :     I                 111    ..
                               I     :     I              111      .
                               I     :     I           111       ..
..                             I     +     I       1111         .
  ...                          I     :     I   1111           ..
     ....                      I     :     1111             ..
         ...                   I     :11111I              ..
            ....               I111111     I           ...
                .....    1111111     :     I        ...
                111111111.     I     :     I    ....
1111111111111111          ......................
===========================================================================
POINT            =    2.000000D+00  S  =   5.0000D-03
DIRECTION        =    1.000000D+00  H  =   3.0000D-02
F0 ( 2.64D-11, 1.42D-02) F1 (-6.11D-02, 1.77D-01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     374

               FIGURE 5 INTERPOLATION TO THE FIRST DERIVATIVE

          INQUIRING  ABOUT  THE VALUE OF THE FIRST DERIVATIVE AT THE CENTER
YIELDS THE NUMBER 5D-4.  THIS IS MUCH LARGER THAN THE VALUE FOR  THE  ERROR
ITSELF,  AND  HENCE  MIGHT  CAUSE  SOME DOUBTS.  HOWEVER, THE DERIVATIVE IS
CALCULATED NUMERICALLY AND HENCE  NOT  EXACTLY.    THE  VALUE  REPORTED  BY
MICROSCOPE  IS  ACTUALLY  THE  TRUNCATION  ERROR.  THIS CAN BE CONFIRMED BY
HALVING THE DISCRETIZATION PARAMETER YIELDING THE DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  26
                               I     :     I                           1111
                               I     :     I                        111 .
                               I     :     I                    1111   .
                               I     :     I                 111     ..
..                             I     :     I             1111       .
  ..                           I     :     I         1111          .
    ..                         I     :     I    11111            ..
      ..                       I     +     I1111               ..
        ...                    I     : 11111                 ..
           ..                  I  11111    I               ..
             ...             11111   :     I             ..
                ...    111111  I     :     I           ..
                1111111        I     :     I       ....
        11111111       ....    I     :     I   ....
11111111                   ....................
===========================================================================
POINT            =    2.000000D+00  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 ( 2.64D-11, 3.03D-03) F1 (-4.21D-02, 7.09D-02)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     418

                                  FIGURE 6

          NOW  THE  VALUE OF THE DERIVATIVE IS 1.25D-4 WHICH IS A REDUCTION
FROM THE PREVIOUS VALUE BY A  FACTOR  4.    THIS  IS  CONSISTENT  WITH  THE
ASSUMPTION  THAT  THE REPORTED ERROR ACTUALLY IS THE TRUNCATION ERROR SINCE
IT DECREASES LIKE THE SQUARE OF THE DISCRETIZATION  PARAMETER  H  (SEE  THE
DISCUSSION IN APPENDIX I).

          THE  FREE  END  CONDITION  (S"(2)  =  0)  CAN  BE  CONSIDERED  AN
INTERPOLATION  CONDITION.    HOWEVER, IT CANNOT BE CHECKED BY EXAMINING THE
ERROR FUNCTION, SINCE THE VALUE OF P"(2) IS (IN GENERAL) UNKNOWN.   SO  ONE
HAS TO CONSIDER THE INTERPOLANT ITSELF.  WE OBTAIN THE DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  27
11                             I     :     I                          22222
  1                            I     :     I                     22222  1
   1                           I     :     I                22222      1
    11                         I     :     I           22222         11
      1                        I     :     I      22222             1
       11                      I     :     I 22222                11
         11                    I     :  22222                   11
           1                   I   22+22   I                   1
            11                22222  :     I                 11
              11         22222 I     :     I               11
                111 22222      I     :     I            111
               222221          I     :     I          11
          22222      111       I     :     I       111
     22222              1111   I     :     I   1111
22222                       1111111111111111111
===========================================================================
POINT            =    0.000000D+00  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 ( 8.83D-01, 1.12D+00) F1 ( 1.26D+00, 1.27D+00) F2 (-2.53D-01, 2.53D-01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     505

                        FIGURE 7 FREE END CONDITION

          THAT THE SECOND DERIVATIVE IS INDEED ZERO  CAN  NOW  BE  VERIFIED
USING  THE  SAME  TECHNIQUES  AS ABOVE (I.E.  INQUIRING DIRECTLY OR USING A
DECREASED DISCRETIZATION PARAMETER).

                              2.2  SMOOTHNESS

          IN  THE  PRESENT  EXAMPLE,  THE  ONLY  CRITICAL  SET  IS  THE ONE

CONTAINING THE POINT X = 1 (ASSUMING WE  ARE  CONFIDENT  THAT  S  HAS  BEEN
PROGRAMMED  TO  BE POLYNOMIAL ELSEWHERE).  WHEN INVESTIGATING SMOOTHNESS IT
IS USUALLY PREFERABLE TO EXAMINE AN  ERROR  (IF  A  PRIMITIVE  FUNCTION  IS
AVAILABLE) RATHER THAN THE INTERPOLANT ITSELF.  IN THE PRESENT EXAMPLE, THE
ERROR FUNCTION AT THE SINGLE CRITICAL POINT YIELDS THE DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  28
                               I     :   3333333333333333333333333333333333
                               I     :  3  I                     .2222
                               I     : 3   I                 12222
                               I     :     I            112222
                               I     :     I       .112222
                               I     :3    I  .1112222
                               I     :    11122222
                               I     +1112222
                               I111112222  I
                          .1111222223:     I
                     1111222222I     :     I
                112222222      I     :     I
           1222222             I   3 :     I
     1222222                   I  3  :     I
3333333333333333333333333333333333   :     I
===========================================================================
POINT            =    1.000000D+00  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 ( 2.49D+00, 2.97D+00) F1 ( 2.39D+00, 2.90D+00) F2 ( 2.48D+00, 3.11D+00)
F3 ( 2.73D+00, 4.06D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     592

                        FIGURE 8 SOOTHNESS AT X = 1

          THIS  DISPLAY  CAN  BE  INTERPRETED  SIMILARLY  AS  FIGURE 1, AND
CLEARLY SHOWS A FUNCTION THAT IS TWICE BUT NOT THREE TIMES  DIFFERENTIABLE.
THIS IS WHAT WE EXPECT FROM OUR CONSTRUCTION.

                          2.3  DEGREE OF PRECISION

          THE 'PRECISION CLASS' OF AN APPROXIMATION SCHEME IS  THE  SET  OF
FUNCTIONS  THAT  ARE  REPRODUCED  EXACTLY  BY  THE  SCHEME.   OF PARTICULAR
INTEREST ARE THE POLYNOMIALS IN THE PRECISION CLASS.  THE LARGEST NUMBER Q,
SAY, SUCH THAT ALL POLYNOMIALS OF DEGREE UP TO Q ARE IN THE PRECISION CLASS
IS THE DEGREE OF POLYNOMIAL PRECISION.

          TO  DECIDE  IF  ANY PARTICULAR FUNCTION IS IN THE PRECISION CLASS
ONE APPROXIMATES THAT FUNCTION AND INVESTIGATES IF THE ERROR IS ZERO WITHIN
ROUND-OFF EFFECTS.  THIS APPLIES EVEN TO NON-POLYNOMIAL FUNCTIONS.

          TO ANSWER THE MORE NARROW QUESTION OF POLYNOMIAL PRECISION IT  IS
POSSIBLE  TO  PROCEED  MORE  SYSTEMATICALLY  AND  TO EXAMINE POLYNOMIALS OF
INCREASING DEGREE.  THE POLYNOMIALS SHOULD BE AS GENERAL AS POSSIBLE,  I.E.
THEY SHOULD BE UNLIKELY TO INTRODUCE MISLEADING ARTIFACTS.  ONE POSSIBILITY
IS TO CHOOSE THE COEFFICIENTS RANDOMLY.  IF  REALLY  IN  DOUBT,  ONE  MIGHT
EXAMINE  EACH  POLYNOMIAL IN A SUITABLE BASIS OF THE APPROPRIATE POLYNOMIAL
SPACE.   THE  POINTS  OF  EXAMINATION  SHOULD  BE  CHOSEN  SUCH  THAT  EACH
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  29
APPROPRIATE  PART  OF  THE DOMAIN IS COVERED.  FOR INSTANCE, IN THE PRESENT
EXAMPLE (WHICH, INCIDENTALLY, HAS LINEAR PRECISION) ONE SHOULD  CHOOSE  ONE
POINT > 1, AND ONE < 1, COVERING BOTH CASES IN THE DEFINITION OF S.

          THE FOLLOWING FIGURE ILLUSTRATES THE TYPICAL APPEARANCE OF A ZERO
ERROR.

                  .      .     I    .:     I  .
                      .    .   I     :     I     .                       .
                          .  . I     :     I                  ..
                               .     :  .  I       .    .           .
                              .I .   :     I      .
                               I   . :     I         .           .        .
    .   .   .                  I  .  .     I.
.               .              I     :     I           .           .
                    .          I     : .   I          .     .     .
                        .      I     :.  . I                            .
                            .  I     :     I    .        .           .
                               I     :     .
 . . . . . . . . .             I     :    .I .             .
                   . .         I.    :     I        .     .     .     ..
  .   .   .   .        .       I     :     I   .             .
===========================================================================
POINT            =    2.342300D-01  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 (-9.99D-11, 9.78D-11)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     667

                           FIGURE 9 A ZERO ERROR

          NOTICE HOW THE VALUES OF THE FUNCTION ARE  SPREAD  RANDOMLY  OVER
THE  ENTIRE  DISPLAY,  ACCURATELY  PINPOINTING  THE ROUND-OFF NATURE OF THE
ERROR.  SOMETIMES, HOWEVER, ONE HAS A SITUATION IN WHICH  THE  COEFFICIENTS
OF THE INTERPOLANT ARE EVALUATED INACCURATELY, LEADING TO AN ERROR THAT HAS
A SCATTERED DISTRIBUTION WITH A BIAS LIKE THAT IN THE FOLLOWING DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  30
                               I     :     I                       . .  .
                               I     :     I                     .  .
                               I     :     I              .    ..         .
                               I     :     I     .           .        .
                               I     :    .I   ..     .    ..     .      .
                               I   ..:     I .         . .             .
                         .    .I     :     ..     .  .  .     .
                      .        I .   :.  . I       .
                       ..      ..    : .   I  .     .
               ..         .  . I  .  :  .  I
 .     ..    .    .  .         I     .     I
.         .        ..      ..  I     :     I
  .  .     ..                  I     :     I
    .         .  .             I     :     I
   .  .  .                     I     :     I
===========================================================================
POINT            =    2.342300D-01  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 ( 2.16D-10, 7.31D-10)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     742

                      FIGURE 10 ROUND-OFF WITH A BIAS

                        2.4  DEGREE OF A POLYNOMIAL

          VERIFYING  THAT  A  CERTAIN FUNCTION IS A POLYNOMIAL OF A CERTAIN
DEGREE IS OFTEN USEFUL WHEN EXAMINING CROSS DERIVATIVES.  FOR  EXAMPLE,  IN
THE  BIVARIATE CLOUGH-TOCHER SCHEME (ALFELD, 1984) WHICH IS PIECEWISE CUBIC
ON A  TRIANGLE,  DIFFERENTIABILITY  BETWEEN  TRIANGLES  IS  FORCED  BY  THE
REQUIREMENT THAT THE NORMAL DERIVATIVES ACROSS EDGES BE LINEAR (RATHER THAN
QUADRATIC) ALONG THE EDGES.  IN THIS ILLUSTRATION, LET US VERIFY  THAT  OUR
INTERPOLANT  IS INDEED CUBIC.  THIS CAN BE ACCOMPLISHED BY SHOWING THAT THE
THIRD DERIVATIVE IS CONSTANT OR THAT THE FOURTH DERIVATIVE IS ZERO  (WITHIN
ROUND-OFF).  ONE OBTAINS FOR EXAMPLE THE FOLLOWING DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  31
  3                            I     :     I                          22222
      3        3               I     :     I                     222221
                  3            I     :     I           3 3 32222211
                              3I     :     I           22222111
 3            3 3              I   3 :  3 3I3     222221111      3       3
       3              33   3   3     :     3 32322 1111           3       3
         3 3 3            3 3  I     :  22222. 11113         3       3
    33                   3   3 I3  23222. 11113   3 3   3   3   3     3 3
                              22222. 11111 I                       3
3       3 3      3  3    22222.I111113 3 3 I                        3
                    22223. 11111 33  :3    I    3     3   3   3        3
   3           22223 311111    I     :     I
          22222111111          I     :     I     3   3         3
     222221111                 I     :     I
2222211     3                  I     :     I
===========================================================================
POINT            =    2.342300D-01  S  =   2.5000D-03
DIRECTION        =    1.000000D+00  H  =   1.5000D-02
F0 ( 1.18D+00, 1.43D+00) F1 ( 1.29D+00, 1.41D+00) F2 ( 3.87D-01, 8.92D-01)
F3 ( 2.73D+00, 2.73D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     841

                         FIGURE 11 A CUBIC FUNCTION

          NOTICE    THAT  THE  THIRD  DERIVATIVE  IS  CLEARLY  AFFECTED  BY
ROUND-OFF,  WITHOUT  ANY  APPARENT  BIAS.    ON  THE  OTHER HAND, SINCE THE
DISPLAYED LIMITS OF ITS RANGE ARE IDENTICAL THE SIZE OF THE RANGE  IS  LESS
THAN  1/273TH  OF  THE DERIVATIVE VALUE.  A CLOSER EXAMINATION REVEALS THAT
THE THIRD DERIVATIVE RANGES FROM 2.7306 TO 2.7323.    A  TELLTALE  SIGN  OF
ROUND-OFF  ERRORS  IS THAT THEY CAN BE DECREASED EVEN FURTHER BY INCREASING
THE DISCRETIZATION PARAMETER.  DOUBLING H YIELDS THE DISPLAY:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  32
                               I     :     I                 3       322222
         3   3                 I 3   :     I                     2222231
33                             I     :     I                22222 111
            3     3 33         I     :  333I           233223  111
        3       3            3 3     :     I    3 22322.  31111
           3          3        I     :     I 22222.     111               3
     3        3           3 3  I   3 33322222.33 33 1111   3     3 3   3
   3   3                 3     I   22222.  I    1111          3     3
    3                  33  3  32322. :     I1111   3 3                  3
                         22222.I  3  : 1111333        3        3
                 3 322222.     I  11311    I                             3
  3       3    22222.        11111   :     I           3          3
          22222.      1111111  I     :     I
     23222.  111111111         I     :     I
2222211111111  3               I     :     I                    3
===========================================================================
POINT            =    2.342300D-01  S  =   5.0000D-03
DIRECTION        =    1.000000D+00  H  =   3.0000D-02
F0 ( 1.06D+00, 1.56D+00) F1 ( 1.27D+00, 1.50D+00) F2 ( 1.34D-01, 1.15D+00)
F3 ( 2.73D+00, 2.73D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     885

                                 FIGURE 12

          THE  RANGE  IS NOW FROM 2.731361 TO 2.731569 WHICH IS A REDUCTION
BY A FACTOR 8.  THIS IS CONSISTENT WITH THE THIRD DERIVATIVE BEING CONSTANT
SINCE  THE ROUND-OFF ERRORS IN THE THIRD DERIVATIVE INCREASE LIKE THE THIRD
POWER OF H, SEE APPENDIX I.  FURTHER  INCREASES  DO  NOT  ALTER  THE  BASIC
SCATTERED STRUCTURE OF THE THIRD DERIVATIVE (UNTIL THE RANGE COVERED IN THE
DISPLAY INCLUDES THE POINT X = 1, WHERE THE THIRD  DERIVATIVE  HAS  A  STEP
DISCONTINUITY).    INCREASING THE VALUE OF H SHOULD EVENTUALLY DECREASE THE
ROUND-OFF ERRORS BELOW A LEVEL AT WHICH THE TRUE CHANGE IN  THE  DERIVATIVE
BECOMES  VISIBLE  IN  THE DISPLAY IF THE THIRD DERIVATIVE WAS NOT CONSTANT.
THE FACT THAT THIS DOES NOT HAPPEN INDICATES THAT THERE IS NO SUCH  CHANGE.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  33
                            3. USING MICROSCOPE

          THIS  SECTION  CONTAINS  A  DESCRIPTION  OF THE MASTER MICROSCOPE
ROUTINE MCRSCP  AND  A  FIRST  INTRODUCTION  TO  THE  COMMAND  LANGUAGE  OF
MICROSCOPE.  FOR A MORE DETAILED DESCRIPTION CONSULT SECTION 4.

          DURING AN INTERACTIVE SESSION THE TERMINAL SCREEN MAY  BE  EITHER
'ACTIVE'  OR  'INACTIVE'.    ON AN ACTIVE SCREEN, CHANGES IN PARAMETERS ARE
INCORPORATED    IMMEDIATELY   INTO  THE  ALPHANUMERICAL  DISPLAY  WHICH  IS
MAINTAINED  ON THE SCREEN.  ON AN INACTIVE SCREEN, COMMANDS ARE ACCUMULATED
AND CARRIED OUT ONLY WHEN THE SCREEN IS ACTIVATED BY EITHER THE COMMAND  GO
OR  FORCE.  AN  INACTIVE  SCREEN  SHOWS  THE  LAST  FEW  COMMANDS  AND  THE
CORRESPONDING DATA WHEREAS AN ACTIVE  SCREEN  SHOWS  AT  MOST  THE  CURRENT
COMMAND AND ITS DATA, IN ADDITION TO THE ALPHANUMERICAL DISPLAY.  (HOWEVER,
AN INCOMPLETE ALPHANUMERICAL DISPLAY MAY BE CALLED ON AN  INACTIVE  SCREEN,
SEE  THE  DESCRIPTION  OF  THE  COMMAND  RSCREEN  IN  SECTION  4.5  AND THE
DISCUSSION OF THE ALPHANUMERICAL DISPLAY IN SECTION 1.3.)

          A  SECOND  DISTINCTION APPLIES TO THE WAY IN WHICH SCREEN CHANGES
ARE IMPLEMENTED ON AN ACTIVE SCREEN.  IN ONE VERSION OF  MICROSCOPE,  WHICH
WE CALL THE 'SCREEN VERSION', THE ALPHANUMERICAL DISPLAY IS KEPT STATIONARY
AND ONLY MODIFIED RATHER THAN REPLACED.  THIS VERSION REQUIRES YOU OR  YOUR
SYSTEM  MANAGER  TO  PROVIDE  FOUR  SIMPLE SCREEN EDITING ROUTINES THAT ARE
SPECIFIC TO EACH COMPUTER/TERMINAL COMBINATION (SEE  CHAPTER  5.)  IF  THIS
FACILITY  IS  NOT  AVAILABLE,  OR  IF YOU USE A HARDCOPY TERMINAL, THEN THE
'SCROLL VERSION' OF MICROSCOPE MAY  BE  EMPLOYED,  WHICH  SCROLLS  THE  OLD
SCREEN AND REPLACES IT BY A NEW VERSION AFTER EACH CHANGE.

          THE SCREEN VERSION IS MUCH PREFERABLE,  AND  IN  THIS  MANUAL  WE
USUALLY ASSUME IT IS AVAILABLE.

          ANOTHER DISTINCTION PERTAINS TO THE  UTILIZATION  OF  LOWER  CASE
LETTERS.    THE  USE  OF  LOWER  CASE  LETTERS  IN A FORTRAN PROGRAM IS NOT
PORTABLE, BUT NOW MOST INSTALLATIONS DO PROVIDE THIS  FACILITY.    IN  THIS
MANUAL  WE  ASSUME LOWER CASE LETTERS ARE AVAILABLE.  IF SO, ANY LOWER CASE
LETTERS OCCURING IN THE INPUT TO THE PROGRAM ARE TREATED AS  IF  THEY  WERE
UPPER CASE.  THUS COMMANDS CAN BE GIVEN IN LOWER CASE LETTERS.  OUTPUT FROM
MICROSCOPE CONSISTS OF A MIXTURE OF LOWER AND UPPER CASE LETTERS.  HOWEVER,
TO  MAINTAIN  PORTABILITY  WE  PROVIDE  A  VERSION OF MICROSCOPE THAT WORKS
SOLELY WITH UPPER CASE LETTERS.  SECTION 5 CONTAINS MORE INFORMATION.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  34
                           3.1 THE MCRSCP ROUTINE

          TO  PUT  MICROSCOPE INTO OPERATION, YOU WILL HAVE HAVE TO WRITE A
FORTRAN MAIN PROGRAM THAT CALLS THE ROUTINE

     MCRSCP(F, INPUT, OUTPUT, GRAPHC, HELP, RECORD, RESTART,
    X       LINES, WIDTH, PLOT, NUMRCL, PROMPT)

          THE  FIRST  PARAMETER  IS THE NAME OF A DOUBLE PRECISION FUNCTION
(THE TRIAL FUNCTION).  IT HAS THE FORM

      DOUBLE PRECISION FUNCTION F(X)
      DOUBLE PRECISION X
      DIMENSION X(1)
              .
              .
              .
      F = ...
      RETURN
      END

AND  MUST BE DECLARED EXTERNAL IN THE CALLING PROGRAM.  MICROSCOPE WILL SET
AT MOST THE FIRST THREE COMPONENTS OF X BEFORE CALLING  F  (I.E.    IT  CAN
HANDLE  ONLY  FUNCTIONS  OF UP TO THREE VARIABLES).  NOTICE THAT F DOES NOT
HAVE ANY PARAMETERS OTHER THAN THE VALUES  OF  THE  INDEPENDENT  VARIABLES.
USUALLY,  HOWEVER,  IT  WILL ALSO DEPEND ON PARAMETERS DESCRIBING E.G.  THE
DOMAIN DATA STRUCTURE.  THESE ARE MOST CONVENIENTLY PASSED TO F IN A COMMON
BLOCK.

          ALL OF THE REMAINING PARAMETERS OF  MCRSCP  ARE  INPUT  INTEGERS.
MCRSCP  WILL  NOT  CHANGE  THEM,  AND  THEY MAY THEREFORE BE INTEGER VALUES
RATHER THAN VARIABLES.  THE FIRST SIX OF THEM (I.E.  INPUT, OUTPUT, GRAPHC,
HELP,  RECORD,  RESTART)  ARE  FORTRAN  DEVICE NUMBERS CORRESPONDING TO THE
FIRST SIX I/O NUMBERS LISTED IN THE NUMERICAL DISPLAY  (SEE  SECTION  1.3).
IT  IS  YOUR RESPONSIBILITY TO SET UP THE NECESSARY CORRESPONDENCES BETWEEN
DEVICE NUMBERS AND PHYSICAL DEVICES OR DISK FILES.

          THE  LAST FIVE PARAMETERS DESCRIBE THE PHYSICAL DIMENSIONS OF THE
SCREEN DISPLAY.  THEIR MEANING IS AS FOLLOWS:  (NUMBERS IN PARENTHESES GIVE
THE VALUE USED FOR THE EXAMPLES IN THIS MANUAL, SQUARE BRACKETS CONTAIN THE
ALLOWABLE RANGES FOR THE PARAMETERS).

          LINES  (24):    THE  TOTAL  NUMBER  OF LINES USED ON THE TERMINAL
SCREEN [10 < LINES < 58].

          WIDTH (75):  THE NUMBER OF COLUMNS USED FOR THE GRAPHICAL DISPLAY
[0 < WIDTH < 136].  USUALLY ONE WILL CHOOSE AS LARGE A  VALUE  AS  POSSIBLE
FOR  MAXIMUM RESOLUTION.  HOWEVER, SMALL VALUES MAY SOMETIMES BE PREFERABLE
FOR INCREASED SPEED.  FOR A SYMMETRICALLY PLACED VERTICAL AXIS,  THE  VALUE
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  35
OF  WIDTH  SHOULD BE ODD.  THE NUMERICAL DISPLAY OCCUPIES UP TO 75 COLUMNS.
THIS IS THEREFORE THE MINIMUM PHYSICAL WIDTH OF THE OUTPUT DEVICE.    WIDTH
IS  ALSO  THE  NUMBER  OF  POINTS  AT  WHICH DERIVATIVES ARE SAMPLED IN THE
<PLOT79> DISPLAY.

          PLOT  (15):    THE  NUMBER  OF  LINES  ALLOCATED TO THE GRAPHICAL
DISPLAY [0 < PLOT < 49].  AGAIN, A VALUE  OF  PLOT  AS  LARGE  AS  POSSIBLE
SHOULD BE SUPPLIED.  THE NUMERICAL EFFORT IS INDEPENDENT OF PLOT.

          NUMRCL (8):  THE NUMBER OF LINES ABOVE THE SCREEN BOTTOM AT WHICH
THE NUMERICAL DISPLAY BEGINS.  [5 + PROMPT < NUMRCL < LINES - PLOT].

          PROMPT (2):  THE NUMBER OF LINES ABOVE THE SCREEN BOTTOM AT WHICH
THE MICROSCOPE PROMPTS ARE GIVEN.  [1 < PROMPT < LINES - PLOT - 6].

          USUALLY, MCRSCP WILL  BE  CALLED  ONLY  ONCE  DURING  A  SESSION.
HOWEVER,  IT IS POSSIBLE TO CALL YOUR OWN ROUTINES FROM MCRSCP. YOU HAVE TO
SUPPLY A SUBROUTINE SUBUSR WITHOUT ANY PARAMETERS WHICH CAN  BE  CALLED  BY
MCRSCP  A DETAILED DISCUSSION OF THIS PROCEDURE IS GIVEN IN SECTION 4.15 ON
USER INTERVENTION.  FOR NOW WE MAY ASSUME THAT SUBUSR IS A DUMMY ROUTINE

      SUBROUTINE SUBUSR
      RETURN
      END

(WHICH  IS SUPPLIED WITH THE MICROSCOPE PACKAGE AND WHICH YOU CAN OF COURSE
WRITE YOURSELF).

          WHEN  CALLED,  MCRSCP  FIRST PRINTS AN IDENTIFYING STENCIL ON THE
OUTPUT DEVICE, CONTAINING ITS VERSION  NUMBER  AND  THE  LAST  MODIFICATION
DATE.    NEXT,  IT  CHECKS  TO  SEE  IF  THE  SCREEN PARAMETERS SATISFY THE
RESTRICTIONS  LISTED  ABOVE.    IF  NOT  THEN  MCRSCP  EXITS,  RATHER  THAN
TERMINATES, IN ORDER TO GIVE YOU A CHANCE TO CORRECT YOUR MISTAKE.

          ON THE FIRST CALL TO MCRSCP IT  COMPUTES  THE  MACHINE  ROUND-OFF
UNIT  (SEE SECTION 4.1) AND PRINTS ITS COMMON LOGARITHM TO GIVE YOU AN IDEA
OF HOW MANY DECIMAL DIGITS ARE CARRIED ON THE COMPUTER.  THE ACTUAL  NUMBER
RELEVANT  TO  YOUR  INVESTIGATION  MAY  OF  COURSE BE LESS DEPENDING ON THE
DETAILS OF THE ARITHMETIC INVOLVED IN EVALUATING THE TRIAL FUNCTION.

          ALSO ON THE FIRST CALL ONLY, MCRSCP THEN PRINTS THE FIRST LINE OF
ANY AVAILABLE NEWS (SEE SECTION 4.10) TO HELP YOU DECIDE  IF  YOU  WISH  TO
READ THEM.

          AFTER SETTING SOME DEFAULTS MCRSCP THEN ENTERS THE COMMAND  MODE,
IN WHICH IT ACCEPTS, INTERPRETS AND CARRIES OUT THE COMMANDS ENTERED ON THE
INPUT DEVICE.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  36
                            3.2 THE COMMAND MODE

          THE MICROSCOPE PROMPT IS OF THE FORM:
 N  >>
WHERE N IS THE NUMBER OF THE COMMAND TO BE GIVEN.  IT STARTS AT  1  AND  IS
INCREMENTED BY 1 AFTER EACH COMMAND.

          A MICROSCOPE COMMAND IS A STRING OF  AT  LEAST  TWO  LETTERS  AND
DIGITS.    HOWEVER,  ONLY  THE  FIRST  TWO  CHARACTERS  ARE SIGNIFICANT AND
RECOGNIZED BY THE PARSER.  THEY MUST BE ENTERED IN THE  FIRST  TWO  COLUMNS
IMMEDIATELY FOLLOWING THE PROMPT.  ON AN ACTIVE SCREEN, THIS WILL BE IN THE
SAME LINE AS THE PROMPT,  ON  AN  INACTIVE  SCREEN,  OR  IN  THE  SCROLLING
VERSION,  ON  THE  NEXT  LINE.   BLANK SPACES ARE SIGNIFICANT.  (MICROSCOPE
SIMPLY READS THE COMMAND USING A 2A1 FORMAT.  THE REMAINING COLUMNS CAN  BE
FILLED  WITH  ARBITRARY  CHARACTERS.    THEY  ARE  NOT READ OR PROCESSED BY
MICROSCOPE.  THIS IS SOMETIMES USEFUL FOR INCLUDING COMMENTS IN A  PROGRAM.
A COMMAND LINE IS TERMINATED BY PRESSING THE RETURN BUTTON.

          AFTER A COMMAND YOU WILL BE PROMPTED FOR THE APPROPRIATE DATA (IF
ANY ARE NEEDED).  THE PROMPT DESCRIBES WHAT DATA ARE NEEDED AND IS SPECIFIC
TO THE COMMAND.  THERE ARE SEVEN TYPES OF DATA REQUESTS:

          -1- NO DATA

          -2- A COMMAND NAME (TO OBTAIN HELP ON THE REQUESTED COMMAND)

          -3- 1 INTEGER

          -4- 2 INTEGERS

          -5- 1 DOUBLE PRECISION VARIABLE

          -6- 1 DOUBLE PRECISION VECTOR

          -7- 2 DOUBLE PRECISION VECTORS

          THE DIMENSION OF THE  VECTORS  IS  IDENTICAL  TO  THE  NUMBER  OF
VARIABLES OF THE TRIAL FUNCTION.  (THE DEFAULT IS 2, SEE SECTION 4.18 FOR A
LISTING OF ALL DEFAULTS).  NUMBERS ARE PROCESSED BY AN INTERNAL PARSER  AND
ARE  ESSENTIALLY  FORMAT FREE.  THE FOLLOWING DISTINCTIONS AND RESTRICTIONS
APPLY, HOWEVER:

          - BLANKS ARE IGNORED.  FOR EXAMPLE "1 2" IS CONSIDERED THE SINGLE
INTEGER 12.

          - NUMBERS ON THE SAME LINE MUST BE SEPARATED BY COMMAS.

          - NUMBERS CAN ALSO BE SEPARATED BY WRITING THEM ON SEPARATE LINES

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  37
(OMITING THE COMMA).

          - IF TWO VECTORS ARE REQUESTED THE SECOND VECTOR MUST START ON  A
NEW LINE .

          - AN EMPTY STRING (OBTAINED BY JUST PUSHING THE RETURN BUTTON) IS
CONSIDERED  TO  MEAN  ZERO  WHENEVER  A  SINGLE NUMBER (INTEGER OR REAL) IS
REQUESTED.  THIS IS PARTICULARLY USEFUL WHEN A ZERO ENTRY  IMPLIES  THAT  A
SUBMODE (SEE THE DESCRIPTIONS OF THE PLOT AND THE USER COMMANDS IN SECTIONS
4.14 AND 4.15 RESPECTIVELY) IS TO BE EXITED.

          -  WHEN  TWO  INTEGERS  ARE  REQUESTED  EITHER ONE OR BOTH CAN BE
REPRESENTED BY AN EMPTY STRING.  (THE ONLY COMMAND FOR WHICH THIS IS USEFUL
IS TYPE.)

          - FLOATING POINT NUMBERS (LIKE 1.0E-12  OR  1D-4)  ARE  PROCESSED
CORRECTLY.

          - FLOATING POINT NUMBERS MAY ALSO BE REPRESENTED AS INTEGERS.

          AFTER  RECEIVING  THE  DATA, MICROSCOPE PROCESSES THE COMMAND AND
THE DATA.  IF THE SCREEN IS ACTIVE THE  COMMAND  IS  CARRIED  OUT  AND  THE
ALPHANUMERICAL  DISPLAY  IS UPDATED, OTHERWISE THE NEXT COMMAND IS PROMPTED
FOR.

          IF  A  COMMAND  IS  NOT  RECOGNIZED  A MESSAGE IS PRINTED TO THAT
EFFECT AND A NEW COMMAND IS PROMPTED FOR.  IF DATA  ARE  REQUESTED,  AND  A
NUMBER  IS  NOT  RECOGNIZED,  THEN NEW DATA WILL BE REQUESTED.  HOWEVER, IF
DATA ARE SYNTACTICALLY  CORRECT,  BUT  OTHERWISE  MEANINGLESS  (E.G.    YOU
REQUEST  THE  GRAPH  OF  THE  117TH  DERIVATIVE),  THEN THE COMMAND WILL BE
ABORTED AND A NEW COMMAND WILL BE PROMPTED FOR.

          THERE ARE CURRENTLY 63 COMMANDS WHICH DIVIDE INTO 11 MAJOR GROUPS
CORRESPONDING TO SECTIONS 4.5 THROUGH 4.15.   RATHER  THAN  DESCRIBING  THE
GROUPS  IN  THIS  SECTION,  WE  ILLUSTRATE  SOME OF THEM IN THE COURSE OF A
SEARCH FOR A POINT OF REDUCED SMOOTHNESS (THE 'ACTIVE POINT'),  GIVING  THE
NAMES  OF  THE  COMMANDS USED, AND FURTHER ILLUSTRATING THE CAPABILITIES OF
MICROSCOPE.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  38
               3.3  A DISCONTINUITY IN THE SECOND DERIVATIVE

          THE TRIAL FUNCTION UNDERLYING THIS EXAMPLE IS

                       2       X                      X
          F(X) = (ETA*X )/2 + E  IF X > 0 AND F(X) = E  OTHERWISE

(WHERE ETA = -0.005).  CLEARLY F IS ONCE BUT NOT  TWICE  DIFFERENTIABLE  AT
THE  ORIGIN.    THE EXPONENTIAL TERM SERVES TO CONCEAL THE DISCONTINUITY IN
THE SECOND DERIVATIVE.  (SIMILAR FUNCTIONS  THAT  ARE  0  THROUGH  5  TIMES
DIFFERENTIABLE  CAN  ALSO  BE  EXAMINED  FOR  ARBITRARY VALUES OF ETA.) THE
PARAMETERS ARE  CHANGED  BY  GIVING  THE  USER  INTERVENTION  COMMAND  USER
(ASSUMING  THE  TEST  PACKAGE HAS BEEN LOADED, SEE APPENDIX III AND SECTION
5).

          LET US PRETEND WE DO NOT KNOW THE THE DEGREE OF SMOOTHNESS BUT WE
WISH TO EXAMINE IT.  WE FIRST INFORM MICROSCOPE THAT THE TRIAL FUNCTION HAS
ONE  INDEPENDENT VARIABLE, USING THE DMNSN COMMAND.  THEN WE ENTER A SEARCH
INTERVAL (HERE [-1,3], SAY) USING THE IINTVL COMMAND.  AT THIS  STAGE,  THE
SCREEN  IS  STILL  INACTIVE.    THIS  IS  CHANGED, AND THE FIRST DISPLAY IS
OBTAINED, BY GIVING THE  GO  COMMAND.    THE  DISPLAY  WILL  REMAIN  ACTIVE
THROUGHOUT THE REMAINDER OF THIS SECTION.  WE OBTAIN:

                               I     :     I                             ..
                               I     :     I                            .
                               I     :     I                          ..
                               I     :     I                         .
                               I     :     I                       ..
                               I     :     I                     ..
                               I     :     I                   ..
                               I     :     I                 ..
                               I     :     I              ...
                               I     :     I           ...
                               I     :     I       ....
                               I     :     I  .....
                               I     : .......
                           ............    I
...........................    I     :     I
===========================================================================
POINT            =    1.000000D+00  S  =   5.4054D-02
DIRECTION        =    4.000000D+00  H  =   3.2432D-01
F0 ( 3.68D-01, 2.01D+01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     960

                                 FIGURE 13

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  39
          THIS DISPLAY SHOWS NO LACK OF SMOOTHNESS AT ALL.  SO, NOT KNOWING
ABOUT  THE  NATURE  OF  THE  SUSPECTED  DISCONTINUITY AT ALL WE TURN ON THE
HORIZONTAL SCALE (USING DSCALE) TO LOCATE ANY  PHENOMENA  OF  INTEREST  AND
ACTIVATE  (USING  DGRAPH)  THE  GRAPH OF THE HIGHEST POSSIBLE (I.E.  SIXTH)
DERIVATIVE.  THESE CHANGES ARE INCORPORATED ONE BY  ONE  (NOT  SHOWN  HERE)
INTO AN ACTIVE SCREEN.  THE FINAL RESULT IS THIS:

                 6             I     :     I                             ..
                               I     :     I                            .
                  6            I     :     I                          ..
                               I     :     I                         .
                      6        I     :     I                       ..
                               I     :     I                     .666666666
                6      6       I     :     I    666666666666666666
66666666666666321.987654666666666666666666666666123456789.123..6789.1234567
              6      6         I     :     I              ...
                               I     :     I           ...
               6               I     :     I       ....
                               I     :     I  .....
                   6           I     : .......
                           ............    I
....................6......    I     :     I
===========================================================================
POINT            =    1.000000D+00  S  =   5.4054D-02
DIRECTION        =    4.000000D+00  H  =   3.2432D-01
F0 ( 3.68D-01, 2.01D+01) F6 (-6.07D+01, 6.27D+01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     972

                                 FIGURE 14

          NOW IT IS CLEAR THAT THERE IS A PHENOMENON OF SOME KIND  OCCURING
AT  THE  POINT  -17  MEASURED  ON  THE  HORIZONTAL  SCALE.  TO EXAMINE THIS
FURTHER, WE SHIFT THE  GRAPH  BY  17  (USING  SHIFT).    THIS  CENTERS  THE
INTERESTING PARTS OF THE GRAPH, YIELDING:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  40
                               I  6  :     I                             ..
                               I     :     I                            .
                               I   6 :     I                          ..
                               I     :     I                         .
                               I     : 6   I                       ..
                               I     :     I                     ..
                               I 6   :  6  I                   ..6666666666
6666666666666666666666666666666654321.12366666666666666666666666689.1234567
                               6     :6    I              ...
                               I     :     I           ...
                               I6    :     I       ....
                               I     :     I  .....
                               I    6: .......
                           ............    I
...........................    I     6     I
===========================================================================
POINT            =    8.108108D-02  S  =   5.4054D-02
DIRECTION        =    4.000000D+00  H  =   3.2432D-01
F0 ( 1.47D-01, 8.00D+00) F6 (-6.07D+01, 6.27D+01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     989

                                 FIGURE 15

          ALREADY AT THIS STAGE IT IS POSSIBLE TO TELL THE  NATURE  OF  THE
REDUCED SMOOTHNESS.  THE SIXTH DERIVATIVE CLEARLY SHOWS FOUR DISTINCT LOCAL
EXTREMA.  THIS QUALIFIES IT AS THE THIRD (NUMERICAL) DERIVATIVE OF A  DELTA
FUNCTION.   THE SECOND DERIVATIVE OF THE TRIAL FUNCTION MUST THEREFORE BE A
STEP    FUNCTION.    THE  TRIAL  FUNCTION  IS  HENCE  ONCE  BUT  NOT  TWICE
DIFFERENTIABLE.    DERIVATIVES OF STEP FUNCTIONS ARE DISCUSSED IN DETAIL IN
SECTION 4.2 AND IN APPENDIX I.

          TO  CONFIRM  OUR  DIAGNOSIS  DIRECTLY  WE  EXAMINE  THE    SECOND
DERIVATIVE.  TURNING ON ITS GRAPH AND ERASING THE DISTRACTING GRAPH OF  THE
TRIAL  FUNCTION (USING EGRAPH), HOWEVER, YIELDS THE FOLLOWING DISPLAY WHICH
SHOWS NO LACK OF SMOOTHNESS AT ALL IN THE SECOND DERIVATIVE:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  41
                               I  6  :     I                             22
                               I     :     I                            2
                               I   6 :     I                          22
                               I     :     I                         2
                               I     : 6   I                       22
                               I     :     I                     22
                               I 6   :  6  I                   226666666666
6666666666666666666666666666666654321.12366666666666666666666666689.1234567
                               6     :6    I              222
                               I     :     I           222
                               I6    :     I       2222
                               I     :     I  22222
                               I    6: 2222222
                           222222222222    I
222222222222222222222222222    I     6     I
===========================================================================
POINT            =    8.108108D-02  S  =   5.4054D-02
DIRECTION        =    4.000000D+00  H  =   3.2432D-01
F2 ( 1.48D-01, 8.08D+00) F6 (-6.07D+01, 6.27D+01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =     989

                                 FIGURE 16

          THE  SECOND  DERIVATIVE APPEARS SMOOTH BECAUSE THE DISCRETIZATION
IS TOO CRUDE TO SHOW ITS DISCONTINUITY.  SO WE KEEP DECREASING THE VALUE OF
THE  DISCRETIZATION PARAMETER (AND SIMULTANEOUSLY THE INTERVAL, KEEPING THE
WINDOW WIDTH FIXED).  SINCE THE ACTIVE POINT IS ONLY APPROXIMATELY  IN  THE
CENTER OF THE DISPLAY, THIS WILL CAUSE IT TO DRIFT TO ONE SIDE WHICH HAS TO
BE COUNTERACTED BY MORE SHIFT COMMANDS.  BECAUSE  OF  THE  ORGANIZATION  OF
MICROSCOPE  IT  IS  PARTICULARLY  EFFICIENT  AND  CONVENIENT  TO DIVIDE (OR
MULTIPLY) WITH POWERS OF TWO.  HALVING H FOUR TIMES (USING HALVE)  FOLLOWED
BY A SHIFT OF 24 AND FOUR MORE HALVINGS YIELDS, AFTER ERASING THE SCALE AND
THE GRAPH OF THE SIXTH DERIVATIVE:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  42
                               I     :     I                           2222
                               I     :     I                       2222
                               I     :     I                    222
                               I     :     I                 222
                               I     :     I              222
                              222222 :     I           222
                            22 I    2:     I       2222
                       2 222   I     2     I    222
                     22 2      I     :22   I 222
                 2222          I     :  22222
             2222              I     :     I
           22                  I     :     I
       2222                    I     :     I
    222                        I     :     I
2222                           I     :     I
===========================================================================
POINT            =   -1.040834D-17  S  =   2.1115D-04
DIRECTION        =    4.000000D+00  H  =   1.2669D-03
F2 ( 9.92D-01, 1.00D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    1353

                                 FIGURE 17

          NOW  IT IS CLEAR THAT THERE IS A JUMP DISCONTINUITY IN THE SECOND
DERIVATIVE OF THE TRIAL FUNCTION.  THE LOCATION OF THE ACTIVE POINT CAN  BE
SEEN  TO  BE  0  +/-  ABOUT  S  = 0.0002.  EVEN THE SIZE OF THE JUMP CAN BE
ESTIMATED.  INQUIRING (USING TYPE) FOR THE VALUES OF THE DERIVATIVE AT  THE
LEFT  AND  RIGHT  END  POINTS OF THE WINDOW (WHERE IN THIS CRUDE DISPLAY IT
ASSUMES    ITS    EXTREMA)  YIELDS  VALUES  OF  9.98846D-1  AND  9.96522D-1
RESPECTIVELY,  CORRESPONDING  TO  A  JUMP OF -2.32D-3.  THE REAL JUMP IS OF
COURSE ETA = -5.0D-3.  THE DISCREPANCY IS DUE TO A  TRUNCATION  ERROR,  NOT
JUST  IN  APPROXIMATING  THE  DERIVATIVE, BUT ALSO IN PICKING THE POINTS OF
EVALUATION.  THE ESTIMATE CAN BE IMPROVED BY EXTRAPOLATING  (LINEARLY)  THE
LEFT  AND RIGHT PARTS OF THE SECOND DERIVATIVE TO THE POINT OF EXAMINATION,
USING AN ESTIMATE OF THE THIRD DERIVATIVE WHICH CAN BE OBTAINED FROM ONE OF
THE  EARLIER  DISPLAYS.    WITH  THE  THIRD DERIVATIVE SET EQUAL TO 1, THIS
PROCEDURE YIELDS AN ESTIMATE OF -2.32D-3 - 12S = -4.86D-3 WHICH IS  A  VERY
GOOD RESULT CONSIDERING THE DIFFICULTIES OF NUMERICAL DIFFERENTIATION.

          IN THE PRESENT SETTING,  IT  IS  NOT  POSSIBLE  TO  DECREASE  THE
DISCRETIZATION  PARAMETER FURTHER WITHOUT SUCCUMBING TO ROUND-OFF ERRORS IN
THE SECOND DERIVATIVE (YOU MAY WISH TO TRY THIS YOURSELF).    HOWEVER,  THE
ABOVE  EXAMPLE  ILLUSTRATES  HOW  TO ALLEVIATE THE LIMITATIONS OF ROUND-OFF
ERRORS.  WE ALREADY RECOGNIZED THE TRUE NATURE OF  THE  REDUCED  SMOOTHNESS
WHEN  EXAMINING  THE  SIXTH  RATHER  THAN  THE SECOND DERIVATIVE.  WITH THE
ACCURACY USED FOR THE EXAMPLES IN THIS MANUAL (10  DIGITS),  STEPS  IN  THE
SECOND DERIVATIVE 300 TIMES AS SMALL AS THE ONE ABOVE CAN BE DETECTED.  THE
FOLLOWING DISPLAY SHOWS THE GRAPH OF THE  SIXTH  DERIVATIVE  FOR  THE  SAME
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  43
FUNCTION  AS ABOVE WITH ETA = -1.7D-5 AND H = 0.24.  THESE VALUES HAVE BEEN
TAKEN FROM TABLE 4 IN APPENDIX I (TAU = 1.0D-10).

                               I     :     I                             66
                               I     :     I                           66
                               I     :     I                         66
                               I     :     I                       66
                               I     :     I                     66
                               I     :     I                  666
                               I     :     I                66
                               I     :     I             666
                               I     :     I         6666
                               I    6:     I     6666
                               I   6 :   6 I 6666
                               I     :  6 666
                              666    6     I
                  666666666666 I 66  : 6   I
666666666666666666             I     :6    I
===========================================================================
POINT            =   -1.040834D-17  S  =   4.0000D-02
DIRECTION        =    4.000000D+00  H  =   2.4000D-01
F6 ( 2.23D-01, 4.40D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    1440

                   FIGURE 18 AT THE LIMITS OF RESOLUTION

          THE SIXTH DERIVATIVE SHOWS 4 EXTREMA, IDENTIFYING THE STEP IN THE
SECOND DERIVATIVE.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  44
             4. A DETAILED DESCRIPTION OF FEATURES AND COMMANDS

                           4.1 ROUND-OFF EFFECTS

          THE  FACT  THAT  ALL  ARITHMETIC  OPERATIONS ARE CARRIED OUT WITH
FINITE    ACCURACY  IS  FUNDAMENTAL  AND  ALL  PERVASIVE  IN  WORKING  WITH
MICROSCOPE.    NUMERICAL  DIFFERENTIATION  IS  PARTICULARLY  SENSITIVE   TO
ROUNDING SINCE IT INVOLVES THE  SUBTRACTION  OF  NEARLY  EQUAL  QUANTITIES,
LEADING  TO  A  CANCELLATION  OF  SIGNIFICANT DIGITS.  CONSEQUENTIALLY, THE
ERROR IN NUMERICAL DIFFERENTIATION DOES NOT DECREASE INDEFINITELY WITH  THE
DISCRETIZATION  PARAMETER, BUT RATHER STARTS TO BEHAVE ERRATICALLY BEYOND A
CERTAIN MINIMUM VALUE AND TENDS TO INCREASE AS THE DISCRETIZATION PARAMETER
IS  DECREASED  FURTHER.    HOWEVER,  IT  IS  ITS VERY RANDOMNESS THAT MAKES
ROUND-OFF ERRORS READILY APPARENT IN A MICROSCOPE DISPLAY SO THAT YOU  WILL
NOT BE MISLED INTO ACCEPTING MEANINGLESS RESULTS AT THEIR FACE VALUE.

          THE PRECISE EFFECT  OF  ROUNDED  ARITHMETIC  DEPENDS  ON  SEVERAL
FACTORS:

          -1- THE NUMBER  OF  SIGNIFICANT  (BINARY)  DIGITS  CARRIED  BY  A
PARTICULAR  COMPUTER.    MOST  USERS  WILL  PREFER  AN APPROXIMATION OF THE
EQUIVALENT NUMBER OF DECIMAL DIGITS, AND SO MICROSCOPE COMPUTES THAT NUMBER
AND  PRINTS  IT  OUT  AT  THE  BEGINNING  OF  A  SESSION.   MORE PRECISELY,
MICROSCOPE COMPUTES THE SMALLEST POWER OF 2, TAU, SAY, SUCH THAT WITHIN THE
GIVEN COMPUTER ARITHMETIC THE QUANTITY 1 + TAU SATISFIES A TEST FOR GREATER
THAN 1, AND THEN PRINTS THE ROUNDED VALUE OF THE NEGATIVE COMMON  LOGARITHM
OF TAU.

          -2-   THE  ACCURACY  WITH  WHICH  THE  TRIAL  FUNCTION  IS  BEING
EVALUATED.   THAT ACCURACY WILL ALWAYS BE LESS THAN OR AT MOST EQUAL TO THE
ACCURACY INDICATED IN -1-.  USUALLY IT WILL BE MUCH LESS, FOR  ANY  OF  THE
FOLLOWING REASONS:

          -2.1- TYPICAL  MULTIVARIATE  INTERPOLANTS  ARE  VERY  COMPLICATED
ALGEBRAICALLY,  INVOLVING  MANY OPERATIONS WHICH TEND TO INCREASE ROUND-OFF
ERRORS.

          -2.2-  IF,  AS  IS FREQUENTLY THE CASE, THE TRIAL FUNCTION IS THE
DIFFERENCE BETWEEN  A  PRIMITIVE  FUNCTION  AND  ITS  INTERPOLANT,  THEN  A
CANCELLATION OF SIGNIFICANT DIGITS OCCURS.

          -2.3- IF THE DISPLAY FUNCTION IS A CROSS DERIVATIVE OF THE  TRIAL
FUNCTION  IT  IS  ALREADY CONTAMINATED BY ROUND-OFF ERRORS DUE TO NUMERICAL
DIFFERENTIATION, THUS EXHIBITING EFFECTS THAT ONE WOULD NORMALLY EXPECT  TO
OCCUR ONLY FOR HIGHER ORDER TANGENTIAL DERIVATIVES.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  45
          -3-  THE  PRECISE  NATURE  OF  ROUND-OFF  EFFECTS  IS  ALSO  VERY
SENSITIVE  TO  THE  RELATIVE  MAGNITUDES  OF VARIOUS TANGENTIAL DERIVATIVES
INVOLVED.  FOR EXAMPLE, A CONSTANT FUNCTION WILL EXHIBIT  ROUND-OFF  ERRORS
AT ANY VALUE OF THE DISCRETIZATION PARAMETER.

          OF THE ABOVE THREE SOURCES OF ROUND-OFF ERRORS, YOU WILL  USUALLY
HAVE A FIRM KNOWLEDGE ONLY OF THE FIRST.  KNOWLEDGE ABOUT THE OTHER TWO MAY
BE GAINED BY USING MICROSCOPE ITSELF.  THIS IS USUALLY MORE EFFECTIVE  THAN
ATTEMPTING A THEORETICAL ANALYSIS.

          THE PRECISE NATURE OF ROUND-OFF ERRORS IS DISCUSSED  IN  APPENDIX
I.  DETAILED  FORMULAS  AND  TABLES ARE GIVEN.  HOWEVER, THE USUAL APPROACH
WILL BE TO EXPERIMENT WITH VARIOUS VALUES OF THE DISCRETIZATION  PARAMETER,
CHOOSING  IT  AS  SMALL  AS POSSIBLE WITHOUT GENERATING THE TELLTALE RANDOM
PATTERN THAT INDICATES ROUND-OFF ERRORS.

                     4.2  DERIVATIVES OF STEP FUNCTIONS

          ONE  PERHAPS  SURPRISING  FEATURE OF NUMERICAL DIFFERENTIATION IS
THAT DOUBTFUL DISCONTINUITIES IN SOME DERIVATIVES CAN  OFTEN  BE  DISPLAYED
VERY  CLEARLY  BY  PLOTTING  HIGHER  ORDER DERIVATIVES, EVEN THOUGH A LARGE
DISCRETIZATION PARAMETER IS  NECESSARY  TO  KEEP  THEM  FREE  OF  ROUND-OFF
ERRORS.  THIS FACT BECOMES QUICKLY APPARENT EMPIRICALLY, AND IS ANALYZED IN
APPENDIX I.

          THE  CONCEPTUAL  BASIS  FOR  THE  APPROACH IS PROVIDED BY A DIRAC
DELTA FUNCTION (SEE COURANT AND HILBERT, 1962, P.456).  A STEP FUNCTION CAN
BE  THOUGHT  OF AS RISING AT THE DISCONTINUITY WITH INFINITE SLOPE FROM ONE
DISCRETE VALUE TO ANOTHER.  IF IT IS PIECEWISE CONSTANT ITS DERIVATIVE  CAN
BE  THOUGHT  OF  AS  A  FUNCTION  THAT  IS  ZERO  EVERYWHERE, EXCEPT AT THE
DISCONTINUITY, WHERE IT  IS  INFINITE  SUCH  THAT  THE  INTEGRAL  OVER  THE
DERIVATIVE  EQUALS  THE  JUMP  IN  THE STEP FUNCTION.  THAT DERIVATIVE IS A
DELTA FUNCTION.

          MORE PRECISELY, A DELTA FUNCTION ABOUT A POINT P, SAY, IS DEFINED
AS THE LIMIT OF A SEQUENCE OF NON-NEGATIVE  FUNCTIONS  THAT  ARE  DIFFERENT
FROM  ZERO  ONLY IN NEIGHBORHOODS OF P. THE DIAMETER OF THOSE NEIGHBORHOODS
TENDS TO ZERO, AND THE VALUE OF THE INTEGRAL  OVER  THE  FUNCTIONS  REMAINS
CONSTANT (AT 1 IN THE CLASSICAL DEFINITION).

          DERIVATIVES OF A DELTA FUNCTION CAN BE VISUALIZED BY  CONSIDERING
A  FUNCTION  IN  THE  ABOVE SEQUENCE.  SUPPOSE IT IS BELL SHAPED.  THEN ITS
DERIVATIVE WILL HAVE TWO LOCAL EXTREMA  OF  OPPOSITE  SIGNS.    THE  SECOND
DERIVATIVE  WILL  HAVE  THREE  EXTREMA,  ETC.  TAKEN TO THE LIMIT, THE K-TH
DERIVATIVE OF A STEP FUNCTION HAS K LOCAL EXTREMA OF ALTERNATING SIGNS, ALL
LOCATED AT THE SAME POINT.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  46
          NUMERICAL  DERIVATIVES  OF  A  STEP  FUNCTION  EXHIBIT  THE  SAME
PHENOMENA,  IF  THE  DISCONTINUITY  IS SMEARED OUT OVER A WINDOW CONTAINING
SUFFICIENTLY MANY POINTS.  THIS IS ILLUSTRATED IN THE PICTURES BELOW.   THE
STEP  DISCONTINUITIES OCCUR IN THE M-TH DERIVATIVES, WHERE M = 0,1, ..., 5.
THE DISCRETIZATION PARAMETER IS H = 1, AND THE SIZE OF THE JUMP IS 1.  MORE
PRECISELY, THE M-TH FUNCTION IS DEFINED BY

                                           M
                  S(X) = 0 IF X < 0   AND X /M! OTHERWISE,

WHERE THE EXCLAMATION MARK DENOTES THE FACTORIAL.

          NOTICE  THAT  AS  THE  DEGREE  OF  THE  DISCONTINUOUS  DERIVATIVE
INCREASES, THE SMOOTHNESS OF THE GRAPHS OF  HIGHER  DERIVATIVES  INCREASES.
THIS  IS CONSISTENT WITH THE INTERPRETATION OF NUMERICAL DIFFERENTIATION AS
A SMOOTHING PROCESS.  (THE NUMERICAL DERIVATIVES HAVE THE  SAME  DEGREE  OF
SMOOTHNESS  AS  THAT OF THE DISPLAY FUNCTION, JUST THE NUMBER OF THE POINTS
OF REDUCED SMOOTHNESS IS INCREASED.)

1111111..........5555555555444444444466666666665555555555222222222211111111
       I                             :                             I
       I                             :                             I
       I         6666666666          :                             I
       I                             :              444444444444444I
       I                             :                             I
       I                             :                   6666666666I
6666666I                             :                             66666666
55555556666666666                    :                             55555555
       444444444444444               :                             I
       I                             :                             I
       5555555555                    :         66666666665555555555I
       I                             :                             I
       I                             :                             I
       333333333333333222226666666666555555555544444333333333333333........
POINT            =    0.000000D+00  S  =   3.3333D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D+00
F0 (-1.00D+00, 9.98D-11) F1 (-5.00D-01, 7.68D-11) F2 (-1.00D+00, 1.00D+00)
F3 (-4.00D+00, 4.00D+00) F4 (-4.80D+01, 4.80D+01) F5 (-2.43D+02, 3.65D+02)
F6 (-7.29D+03, 7.29D+03)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    1662

             FIGURE 19 DERIVATIVES OF A STEP FUNCTION  (M = 0)

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  47
222222222211              665       444        66 33333          2222222222
       I  221111         65 65     4 : 4      6 36     33      22  I
       I    22  1111   665   655  4  :  4    633  66     33  22    I
       I      22    11615       54   :   4  33      6    5 33      I
       I        22   6 51111  6 45   :    436        6  5255533    I
66666666          266 5     1116  55 :  33 6          662    5553  66666666
       I666666    6 25        4 11115:33    4        22 6    666666I
555555555     6666  5 22     4  6   3511  6  4     22 5  6666     555555555
       I 555       5    22  4    633 :551611  4  22  5             I
44444444   355            24    33   :  5   11142   5              44444444
       I444  3555 5       4 2233  6  :  65   22 4115            444I
       I   444 335       4  3322   6 : 6  552    45 1111     444   I
       I      44433     4 33    22   :   22 5    54     11444      I
       I         4443  433        226:622    5  5  4   444  1111   I
       I            4443            262       55    444         11111111111
===========================================================================
POINT            =    0.000000D+00  S  =   3.3333D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D+00
F1 (-1.00D+00, 4.28D-11) F2 (-1.00D+00, 2.85D-10) F3 (-2.00D+00, 2.00D+00)
F4 (-8.00D+00, 1.60D+01) F5 (-8.10D+01, 8.10D+01) F6 (-1.46D+03, 9.72D+02)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    1797

                                           0
               FIGURE 20 DERIVATIVES OF A C  FUNCTION (M = 1)

333333333333322222           666665555555   4444444           3333333333333
       I     33   22222     6    56  :   544       44       33     I
       I       33      2222     5    :   45          4    33       I
       I         3         62  5   6 :  4  5          44 3         I
       I          33      6  252     : 4    5          644         I
       I            3        5  22  6:       5      666 6664       I
       I             3   6        222:4            6 3     6666    I
6666666666666         3 6   5        62       5   6 3          666666666666
555555555   4666       6   5        4: 22      5 6 3              555555555
       I 555   46666666 3            :6  22       3            555 I
       I    55    4      35        4 :     222  63           55    I
       I      55   44    5 3      4  : 6      2625         55      I
       I        5    4      33   4   :       33  222      5        I
       I         5    455     344    :  6  33 6   55222225         I
       I          55555 4444444 33333333366666      55555222222222222222222
===========================================================================
POINT            =    0.000000D+00  S  =   3.3333D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D+00
F2 (-1.00D+00, 1.99D-10) F3 (-1.00D+00, 1.74D-09) F4 (-2.67D+00, 2.67D+00)
F5 (-8.98D+00, 1.35D+01) F6 (-1.46D+02, 1.46D+02)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    1932

                                           1
               FIGURE 21 DERIVATIVES OF A C  FUNCTION (M = 2)
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  48

444444444444444443333               666     5555555       44444444444444444
       I         444 3333          6 : 6   5       5   444         I
       I            4    333      6  :  6 5         554            I
       I             44     33   6   :   6          445            I
       I               4      33     :  5          4   55          I
       I                4       633  : 5  6       4      5         I
       I                 4         33:5          4        555      I
55555555555555            4    6     53    6    4            55555555555555
       I      555          4        5: 33      4                   I
       I         55         4 6    5 :   33 6 4                    I
6666666666666666   5         6    5  :     336             6666666666666666
       I        66  5         4  5   :      433          66        I
       I          66 55     6  45    :    44  6333     66          I
       I            6  5   6  55 4   :   4     6  33336            I
       I             666666555    4444444       666666333333333333333333333
===========================================================================
POINT            =    0.000000D+00  S  =   3.3333D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D+00
F3 (-1.00D+00, 1.60D-09) F4 (-1.33D+00, 1.14D-08) F5 (-3.00D+00, 3.00D+00)
F6 (-1.28D+01, 2.70D+01)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    2067

                                           2
               FIGURE 22 DERIVATIVES OF A C  FUNCTION (M = 3)

555555555555555555544444             :    666666        5555555555555555555
       I           55   4444         :   6      6     55           I
       I             5      44       :  6        66  5             I
       I              5       44     :             65              I
       I               5        44   : 6           566             I
       I                5         4  :6           5   6            I
       I                 5         44:           5     666         I
66666666666666666         5          6          5         66666666666666666
       I         666       5         :44       5                   I
       I            66      5       6:  4     5                    I
       I              6      5     6 :   44  5                     I
       I               6      5   6  :     45                      I
       I                66     55    :    55 444                   I
       I                  6     66   :   5      444                I
       I                   66666  5555555          444444444444444444444444
===========================================================================
POINT            =    0.000000D+00  S  =   3.3333D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D+00
F4 (-1.00D+00, 9.60D-09) F5 (-1.38D+00, 6.61D-08) F6 (-4.13D+00, 4.13D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    2202

                                           3
               FIGURE 23 DERIVATIVES OF A C  FUNCTION (M = 4)

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  49
6666666666666666666665555            :                666666666666666666666
       I             66  555         :              66             I
       I               66   55       :            66               I
       I                 6    55     :           6                 I
       I                  6     55   :          6                  I
       I                   6      5  :         6                   I
       I                           55:                             I
       I                    6        5        6                    I
       I                     6       :55     6                     I
       I                      6      :  5   6                      I
       I                       6     :   556                       I
       I                        6    :    655                      I
       I                         6   :   6   55                    I
       I                          6  :  6      555                 I
       I                           66666          5555555555555555555555555
===========================================================================
POINT            =    0.000000D+00  S  =   3.3333D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D+00
F5 (-1.00D+00, 5.09D-08) F6 (-1.65D+00, 2.67D-06)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    2337

                                           4
               FIGURE 24 DERIVATIVES OF A C  FUNCTION (M = 5)

                              4.3  THE WINDOW

          SOMETIMES IT IS DESIRABLE TO USE VERY SMALL WINDOW WIDTHS OR EVEN
COMPLETELY SEPARATE DIFFERENTIATION STENCILS, FOR EXAMPLE WHEN SEARCHING  A
LARGE DOMAIN FOR A POINT OF REDUCED SMOOTHNESS.  THEN STEPS IN A DERIVATIVE
WILL APPEAR AS SUCH RATHER THAN AS GRADUAL SLOPES THAT MIGHT BE OBSCURED BY
OTHER  FEATURES  OF  THE  RELEVANT  GRAPHS.  ALSO, CUSPS AND POLES ARE BEST
DISPLAYED USING  SMALL  WINDOWS  (SEE  APPENDIX  III  FOR  SOME  EXAMPLES).
USUALLY, HOWEVER, IT WILL BE PREFERABLE TO USE LARGER WINDOWS.

          THERE ARE TWO MAIN REASONS  TO  USE  OVERLAPPING  DIFFERENTIATION
STENCILS:    FIRST,  AS  ILLUSTRATED  IN  THE PRECEDING SUBSECTION, THIS IS
NECESSARY TO RESOLVE ALL THE EXTREMA OF A DERIVATIVE OF A  DELTA  FUNCTION,
AND,  SECOND,  IT  ALLOWS THE USE OF ONE EVALUATION OF A TRIAL FUNCTION FOR
SEVERAL    VALUES  OF  A  DERIVATIVE,  THEREBY  INCREASING  EFFICIENCY  AND
ACCELERATING  YOUR INTERACTION WITH MICROSCOPE.  USING OVERLAPPING STENCILS
IS EQUIVALENT TO HAVING A WINDOW WIDTH (W = 2H/S) THAT IS GREATER THAN 1.

          DUE  TO  THE  INTERNAL  ORGANIZATION OF MICROSCOPE, FOR ANY GIVEN
WINDOW WIDTH, THE NUMBER OF EVALUATIONS (OF THE DISPLAY FUNCTION)  REQUIRED
IS DETERMINED SOLELY BY THE LARGEST DEGREE OF THE PLOTTED DERIVATIVES.  THE
FOLLOWING TABLE LISTS FORMULAS FOR COMPUTING THE NUMBER  N  OF  EVALUATIONS

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  50
REQUIRED  GIVEN  THE  WINDOW  WIDTH  (W),  THE  HIGHEST DEGREE OF A PLOTTED
DERIVATIVE (K) AND THE NUMBER OF COLUMNS (C) IN THE GRAPHICAL DISPLAY.

    MAX K:     0        1 OR 2          3 OR 4          5 OR 6

 W=2H/S

 < 1           C          3C              5C              9C

  2            C         C + 2          2C + 3          4C + 5

  4            C         C + 4           C + 4          3C + 8

  6            C         C + 6          2C + 9          2C + 9

  8            C         C + 8           C + 9          3C + 16

 10            C         C + 10         2C + 15         4C + 25

 12            C         C + 12          C + 12          C + 12

 14            C         C + 14         2C + 21         4C + 35

 16            C         C + 16          C + 16         3C + 32

 18            C         C + 18         2C + 27         2C + 27

 20            C         C + 20          C + 20         3C + 40

 22            C         C + 22         2C + 33         4C + 55

 24            C         C + 24          C + 24          C + 24

                    TABLE:  EFFICIENCY OF WINDOW WIDTHS

          THE  SPECIAL  ROLE  PLAYED  BY  W = 12 IS OBVIOUS:  THE NUMERICAL
EFFORT IS INDEPENDENT OF THE DEGREE OF THE PLOTTED DERIVATIVES, AND EXCEEDS
THE  NUMBER  OF  COLUMNS  IN  THE DISPLAY ONLY BY A CONSTANT.  THIS IS TRUE
WHENEVER W IS A MULTIPLE OF 12.  THE REASON FOR THIS IS,  OF  COURSE,  THAT
THE  DISPLAY  FUNCTION  IS  SAMPLED  AT  MULTIPLES  OF  1/2  AND 1/3 OF THE
DISCRETIZATION PARAMETER.  FOR EFFICIENCY, THE DEFAULT VALUE OF  W  IS  12.
THIS  IS  ALSO  SUFFICIENT FOR MOST EXAMINATIONS INVOLVING DELTA FUNCTIONS.
SOMETIMES, HOWEVER, A LARGER MULTIPLE, LIKE 24 OR 36 MAY BE DESIRABLE.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  51
                      4.4  REFERENCE TABLE OF COMMANDS

          THIS SUBSECTION CONTAINS A QUICK REFERENCE TABLE OF ALL AVAILABLE
COMMANDS.  FOR EACH COMMAND, A ONE-LINE DESCRIPTION IS GIVEN.   OFTEN,  THE
INFORMATION  CONTAINED  IN  THESE  SHORT  DESCRIPTIONS  WILL BE ALL THAT IS
NEEDED.  THEREFORE, THE DESCRIPTIONS ARE USED IN SEVERAL  OTHER  PLACES  AN
APPROPRIATE  SELECTION  OF  THEM  OCCURS  AT  THE  BEGINNING OF EACH OF THE
FOLLOWING SUBSECTIONS.  THE HSUMRY COMMAND LISTS THEM  ORDERED  BY  TOPICS.
THE  MORE  DETAILED  HELP  COMMAND  GIVES  A DOCUMENTATION OF AN INDIVIDUAL
COMMAND HEADED BY THE ONE-LINE DESCRIPTION.

NAME                DESCRIPTION                 DATA   ERROR   SCREEN

ACCENT  MARK DERIVATIVE WITH ASTERISKS  TOGGLE   K    -1<K<7   IMMEDIATE
C1CROSS READ 1ST COMPONENT OF CROSS DIRECTION    C1     C><0   DEACTIV.
C2CROSS READ 2ND COMPONENT OF CROSS DIRECTION    C2    C><0;1  DEACTIV.
C3CROSS READ 3RD COMPONENT OF CROSS DIRECTION    C3    C><0;1  DEACTIV.
CCHANNL CHANGE CHANNEL NUMBER                    IC,NC  4        ---
CDIRCTN READ DIRECTION OF CROSS DIRECTION        C      C><0   DEACTIV.
CHVALUE READ CH FOR CROSS DIFFERENTIATION        CH     5      DEACTIV.
CORDER  READ ORDER OF CROSS DERIVATIVE           K     -1<K<7  DEACTIV.
CWINDOW CHANGE WINDOW WIDTH BY FACTOR     SIGN   M      M=0    IMMEDIATE
D1CHDIR READ 1ST COMPONENT OF DIR. OF INV.       D1     D><0   DEACTIV.
D2CHDIR READ 2ND COMPONENT OF DIR. OF INV.       D2    D><0;1  DEACTIV.
D3CHDIR READ 3RD COMPONENT OF DIR. OF INV.       D3    D><0;1  DEACTIV.
DCENTER DRAW CENTER OF GRAPH. DISPLAY   TOGGLE   ---    ---    IMMEDIATE
DGRAPH  DRAW GRAPH                               K     -1<K<7  IMMEDIATE
DIVIDE  DIVIDE H BY INTEGER M             SIGN   M      D><0   IMMEDIATE
DMNSN   READ DIMENSION V OF DOMAIN               V     0<V<4   DEACTIV.
DOUBLE  DOUBLE H                                 ---    ---    IMMEDIATE
DSCALE  DRAW SCALE IN GRAPHICAL DISPLAY TOGGLE   ---    ---    IMMEDIATE
DXAXIS  DRAW HORIZONTAL AXIS            TOGGLE   ---    ---    IMMEDIATE
EGRAPH  ERASE GRAPH                              K     -1<K<7  IMMEDIATE
EXIT    RETURN TO CALLING PROGRAM                ---    ---    DEACTIV.
FLIP    REPLACE D BY -D                 TOGGLE   ---    ---    IMMEDIATE
FORCE   RECOMPUTE EVERYTHING                     ---    ---    ACTIV
GO      COMPUTE WHATEVER IS NEEDED               ---    ---    ACTIV
HALVE   HALVE H                                  ---    ---    IMMEDIATE
HELP    OBTAIN DETAILED HELP ON COMMAND CMD      CMD     2       ---
HSUMRY  OBTAIN A SUMMARY OF ALL COMMANDS         ---    ---      ---
IDIRCTN READ THE DIRECTION OF INVESTIGATION      D      D><0   DEACTIV.
IHVALUE READ VALUE OF H                          H      H>0    DEACTIV.
IINTVL  READ ENDPOINTS OF LINE OF INVESTIG.      A,B    A><B   DEACTIV.
IPOINT  READ THE NEW POINT OF EXAMINATION        P      ---    DEACTIV.
LCROSS  LOAD A NEW CROSS DIRECTION               KC      3     DEACTIV
LDRCTN  LOAD A NEW DIRECTION OF INVESTIGATION    KD      3     DEACTIV
LIST    PRINT ALL CURRENTLY AVAILABLE COMMANDS   ---    ---      ---
LOG     GIVE CHANNEL NUMBER FOR LOGGING          NC     ---      ---
LPOINT  LOAD A NEW POINT OF EXAMINATION          KP      3     DEACTIV.
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  52
MULTPLY MULTIPLY H BY FACTOR M            SIGN   M      M><0   IMMEDIATE
NEWS    PRINT THE NEWS                           ---    ---      ---
NORMAL  TURN NORMALIZATION OF D ON/OFF  TOGGLE   ---    ---    DEACTIV.
OUTPUT  WRITE SCREEN IMMAGE ON RECORDING DEVICE  ---    ---      ---
P1CHPNT READ FIRST COMPONENT OF P                 P1    ---    DEACTIV.
P2CHPNT READ SECOND COMPONENT OF P                P2     1     DEACTIV.
P3CHPNT READ THIRD COMPONENT OF P                 P3     1     DEACTIV.
PAUSE   PAUSE UNTIL RETURN FROM CHANNEL NC        NC    ---      ---
PLOT    ENTER PLOTTING (<PLOT79>) MODE           ---    ---      ---
QUIT    TERMINATE MICROSCOPE SESSION             ---    ---      ---
RCROSS  READ CHANNEL NUMBER FOR CROSS LIST        NC    ---      ---
RDIRCTN READ CHANNEL NUMBER FOR DIRECTION LIST    NC    ---      ---
RESTART REESTABLISH EARLIER PARAMETER SETTINGS   ---    ---      ---
ROTATE  CHANGE DIRECTION OF INVESTIGATION         DD    D><0   DEACTIV.
RPOINT  READ CHANNEL NUMBER FOR POINT LIST        NC     1       ---
RSCREEN REFRESH SCREEN                           ---    ---    PENDING
RWIND   REWIND DEVICE NC                          NC    ---      ---
SETDF   RESET DEFAULTS                           ---    ---    DEACTIV.
SHIFT   SHIFT POINT OF EXAMINATION                IS    ---    IMMEDIATE
STORE   STORE CURRENT PARAMETER SETTINGS         ---    ---      ---
TCENTER TYPE VALUE OF K-TH DERIVATIVE AT CENTER   K    -1<K<7    ---
TNOTE   TAKE A NOTE ON RECORDING DEVICE          ---    ---      ---
TYPE    TYPE VALUE OF DERIVATIVE AT SOME POINT   K,IS  -1<K<7  IMMEDIATE
UNDO    UNDO PREVIOUS PARAMETER CHANGES          ---    ---      ---
USER    ENTER USER MODE                          ---    ---      ---
WAIT    MAKE PROGRAM WAIT FOR GO OR FORCE COM.   ---    ---    DEACTIV.
ZOOM    CHANGE WINDOW WIDTH BY FACTOR      SIGN   M     M><0   IMMEDIATE

                     QUICK REFERENCE TABLE OF COMMANDS

                         KEY TO TABLE OF COMMANDS:

                             --  DESCRIPTION --

TOGGLE         THE COMMAND REVERSES ITSELF IF CALLED AGAIN

SIGN           MEANS  THAT  REVERSING  THE  SIGN  OF  THE  INPUT  PARAMETER
               GENERATES  THE  OPPOSITE  EFFECT  (E.G.  MULTIPLY INSTEAD OF
               DIVIDE).

                               --   DATA  --

A              ONE ENDPOINT OF THE LINE OF INVESTIGATION
B              THE OTHER ENDPOINT OF THE LINE OF INVESTIGATION (P  =  (A  +
               B)/2)
C              CROSS DIRECTION
C1             FIRST COMPONENT OF CROSS DIRECTION
C2             SECOND COMPONENT OF CROSS DIRECTION
 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  53

C3             THIRD COMPONENT OF CROSS DIRECTION
CH             DISCRETIZATION PARAMETER FOR CROSS DERIVATIVES
CMD            COMMAND NAME
D              DIRECTION OF INVESTIGATION
D1             FIRST COMPONENT OF DIRECTION OF INVESTIGATION
D2             SECOND COMPONENT OF DIRECTION OF INVESTIGATION
D3             THIRD COMPONENT OF DIRECTION OF INVESTIGATION
DD             CORRECTION TO BE ADDED TO D
IC             TYPE  OF  DEVICE  (1  =  INPUT, 2 = OUTPUT, 3 = GRAPHIC, 4 =
               RECORD, 5 = RESTART)
IS             INTEGER  SHIFT  IN  NUMBER OF COLUMNS ON DISPLAY, +VE SHIFTS
               GRAPH TO RIGHT, POINT TO LEFT, -VE IS REVERSE
K              DEGREE OF A DERIVATIVE (TO BE PLOTTED, ERASED, ETC.)
KC             POSITION OF THE CROSS DIRECTION IN A LIST
KD             POSITION OF THE DIRECTION OF INVESTIGATION IN A LIST
KP             POSITION OF THE POINT OF EXAMINATION IN A LIST
M              INTEGER MULTIPLIER.
N              INTEGER DIVIDER
NC             CHANNEL (DEVICE) NUMBER
P              POINT OF EXAMINATION
P1             FIRST COMPONENT OF THE POINT OF EXAMINATION
P2             SECOND COMPONENT OF THE POINT OF EXAMINATION
P3             THIRD COMPONENT OF THE POINT OF EXAMINATION
---            NO DATA ARE REQUIRED

                               --  ERROR  --

          RESTRICTIONS  ON  THE  DATA ARE LISTED IN THIS COLUMN.  THE ENTRY
"---" MEANS THAT NO CHECK IS MADE FOR ERRORS.  THE SYMBOL "><"  MEANS  "NOT
EQUAL".    IF  THE  ERROR  CONDITIONS  CANNOT  BE GIVEN EXPLICITLY THEN THE
FOLLOWING REFERENCE NUMBERS HAVE BEEN USED:

          1.    COMPONENTS  TO  BE  CHANGED  MUST NOT HAVE AN INDEX THAT IS
GREATER THAN THE CURRENT DIMENSION.

          2.  THE COMMAND MUST BE A VALID MICROSCOPE COMMAND.  IF IT IS NOT
RECOGNIZED YOU WILL BE PROMPTED  FOR  ANOTHER  COMMAND  TO  BE  HELPED  ON.
TYPING  ANY VALID COMMAND WILL GET YOU SOME INFORMATION ON IT AND WILL THEN
RETURN YOU TO COMMAND MODE.

          3.   THERE MUST BE ONE VECTOR IN LEGAL FORMAT IN EACH LINE OF THE
LIST.  IF ILLEGAL CHARACTERS ARE ENCOUNTERED, AN ERROR TERMINATION  OF  THE
COMMAND  WILL  OCCUR.  ALSO, THE INDEX NUMBER MUST NOT EXCEED THE NUMBER OF
LINES IN THE LIST.  IF IT DOES, THE PROGRAM WILL REACH THE END OF THE  FILE
CONTAINING  THE  LIST,  AND  THE PRECISE ACTION AFTER THAT DEPENDS UPON THE
COMPUTING SYSTEM.

                               --  SCREEN  --

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  54
ACTIV          THE COMMAND ACTIVATES THE SCREEN

DEACTIV.       THE COMMAND DEACTIVATES THE SCREEN - I.E. THE  GO  OR  FORCE
               COMMAND  MUST  BE GIVEN BEFORE THE COMPUTATION PROCEEDS. FOR
               SOME COMMANDS, THERE IS A CHECK IF THE NEWLY INPUT DATUM  IS
               IDENTICAL WITH THE OLD ONE, IN WHICH CASE THE SCREEN REMAINS
               ACTIVE.

IMMEDIATE      IF  THE SCREEN IS ACTIVE THEN THE EFFECTS OF THE COMMAND ARE
               INCORPORATED BEFORE THE NEXT COMMAND IS REQUESTED

PENDING        IF  THE  SCREEN  IS  INACTIVE,  THIS  COMMAND  WILL  PUT THE
               GRAPHICAL DISPLAY ON THE SCREEN, EVEN  IF  COMPUTATIONS  ARE
               PENDING.

---            THE GRAPHICAL AND NUMERICAL DISPLAY, AND THE  SCREEN  STATUS
               ARE  NOT  CHANGED.  NOTICE,  HOWEVER, THAT THE ACTUAL SCREEN
               DISPLAY    MAY  BE  DESTROYED  BY  ERROR  MESSAGES  OR  HELP
               INFORMATION.  THE NEXT DISPLAY CHANGING COMMAND (OR RSCREEN)
               WILL THEN REFRESH THE SCREEN.

                    4.5  COMMANDS TO CONTROL THE DISPLAY

          IN THIS AND THE FOLLOWING SUBSECTIONS, INPUT TO MICROSCOPE IS VIA
THE ASSIGNED INPUT CHANNEL, UNLESS OTHERWISE STATED.  THIS WILL USUALLY  BE
THE TERMINAL, BUT MAY ALSO BE A FILE FOR A BATCH TYPE SESSION.

ACCENT  MARK DERIVATIVE WITH ASTERISKS  TOGGLE   K    -1<K<7   IMMEDIATE
DCENTER DRAW CENTER OF GRAPH. DISPLAY   TOGGLE   ---    ---    IMMEDIATE
DGRAPH  DRAW GRAPH                               K     -1<K<7  IMMEDIATE
DMNSN   READ DIMENSION V OF DOMAIN               V     0<V<4   DEACTIV.
DSCALE  DRAW SCALE IN GRAPHICAL DISPLAY TOGGLE   ---    ---    IMMEDIATE
DXAXIS  DRAW HORIZONTAL AXIS            TOGGLE   ---    ---    IMMEDIATE
EGRAPH  ERASE GRAPH                              K     -1<K<7  IMMEDIATE
FLIP    REPLACE D BY -D                 TOGGLE   ---    ---    IMMEDIATE
RSCREEN REFRESH SCREEN                           ---    ---    PENDING
TCENTER TYPE VALUE OF K-TH DERIVATIVE AT CENTER   K    -1<K<7    ---
TYPE    TYPE VALUE OF DERIVATIVE AT SOME POINT   K,IS  -1<K<7  IMMEDIATE

          WE NOW TURN TO THE DESCRIPTIONS OF INDIVIDUAL COMMANDS.    SCREEN
DISPLAY  COMMANDS  CONTROL  WHAT  APPEARS ON THE TERMINAL SCREEN.  (FOR THE
EFFECTS OF THESE COMMANDS ON THE <PLOT79> DRAWING SEE SECTION 4.14.)  THERE
ARE SEVERAL SUBGROUPS OF COMMANDS:

        -1- DECIDING ON THE TANGENTIAL DERIVATIVES TO BE DISPLAYED.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  55
          BY DEFAULT, ONLY THE DISPLAY FUNCTION IS DISPLAYED.  THE GRAPH OF
THE  DISPLAY  FUNCTION  ITSELF  IS MARKED BY PERIODS, THE GRAPH OF ITS K-TH
TANGENTIAL DERIVATIVE (K = 1,2, ..., 6) BY THE DIGIT K.   USING  DGRAPH  OR
EGRAPH  RESPECTIVELY,  YOU  CAN  TURN  ON  OR  OFF THE GRAPHS OF THE ZEROTH
THROUGH SIXTH DERIVATIVES.  IT IS ALSO POSSIBLE TO DISPLAY NO FUNCTIONS  AT
ALL.    IN  ANY  CASE, THE DISPLAY FUNCTION IS EVALUATED ONLY AT THE POINTS
WHERE IT IS NEEDED.

                         -2- ACCENTUATING A GRAPH.

          IF  YOU  WISH  TO  EMPHASIZE THE GRAPH OF A PARTICULAR DERIVATIVE
THEN YOU CAN PLOT ITS GRAPH USING ASTERISKS RATHER THAN  A  PERIOD  OR  THE
DIGIT  K.    THE COMMAND ACCENT TURNS THIS OPTION ON OR OFF.  NOTE THAT THE
NUMERICAL DISPLAY CONTAINS NO DIRECT INFORMATION ON WHICH DERIVATIVES  HAVE
BEEN  ACCENTUATED.    ALSO,  IF  YOU ACCENTUATE SEVERAL DERIVATIVES AT ONCE
THEIR GRAPHS CANNOT BE DISTINGUISHED FROM EACH OTHER.    THE  GRAPH  OF  AN
ACCENTED DERIVATIVE OVERWRITES ANYTHING ELSE IN THE DISPLAY, EXCEPT FOR THE
CENTER MARK.  THE ACCENT COMMAND ALSO HAS THE EFFECT OF TURNING  THE  GRAPH
OF  A  DERIVATIVE  ON,  I.E.    IT DOES NOT NEED TO BE PRECEDED BY A DGRAPH
COMMAND.  BY DEFAULT NO DERIVATIVE IS ACCENTED.

                       -3- DRAWING A HORIZONTAL AXIS

          A  HORIZONTAL  AXIS  CAN  BE  DRAWN  THROUGH  THE  CENTER  OF THE
GRAPHICAL DISPLAY, USING THE COMMAND DXAXIS. IF DESIRED,  A  SCALE  CAN  BE
OBTAINED  BY USING DSCALE. BOTH COMMANDS ARE ON/OFF SWITCHES.  THE POINT AT
WHICH THE AXIS OR SCALE INTERSECTS THE  CENTRAL  COLUMN  OF  THE  GRAPHICAL
DISPLAY  DOES  NOT IN GENERAL CORRESPOND TO THE VALUE ZERO.  (THAT POINT IS
DIFFERENT FOR EACH OF THE TANGENTIAL DERIVATIVES PLOTTED, AND MAY NOT  EVEN
BE  CONTAINED IN THE GRAPHICAL DISPLAY.) THE PURPOSE OF THE AXIS IS TO GIVE
HELP IN EXAMINING SYMMETRY PROPERTIES, AND THE SCALE IS USEFUL IN  LOCATING
POINTS  OF  INTEREST FOR MORE DETAILED EXAMINATION.  BOTH THE SCALE AND THE
AXIS ARE OFF BY DEFAULT.

                           -4- MARKING THE CENTER

          THE  ON/OFF COMMAND DCENTER MAY BE USED TO MARK THE CENTER OF THE
GRAPHICAL DISPLAY  WITH  A  PLUS  SIGN.    THIS  IS  SOMETIMES  USEFUL  FOR
CONFIRMING SYMMETRY PROPERTIES.  THE CENTER MARK IS OFF BY DEFAULT.

                -5- FLIPPING THE DIRECTION OF INVESTIGATION

          THE FLIP COMMAND HAS THE EFFECT OF  REPLACING  THE  DIRECTION  OF
INVESTIGATION  D  BY  -D.  THIS TURNS THE GRAPH OF ANY DERIVATIVE ABOUT THE
VERTICAL AXIS, AND IT ALSO TURNS THE GRAPH OF  ANY  ODD  DEGREE  DERIVATIVE
ABOUT  THE HORIZONTAL AXIS, THEREBY POSSIBLY DISENTANGLING GRAPHS THAT HAVE
BEEN OVERWRITTEN BY OTHERS.

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  56
          CONSIDER  FOR  EXAMPLE  THE  EXPONENTIAL  FUNCTION  AND ITS FIRST
DERIVATIVE.  ORDINARILY THE TWO  GRAPHS  WOULD  BE  IDENTICAL,  YIELDING  A
DISPLAY IN WHICH THE DISPLAY FUNCTION IS COMPLETELY COVERED BY THE GRAPH OF
ITS FIRST DERIVATIVE:

                               I     :     I                            111
                               I     :     I                         111
                               I     :     I                     1111
                               I     :     I                  111
                               I     :     I              1111
                               I     :     I          1111
                               I     :     I      1111
                               I     :     I  1111
                               I     :   11111
                               I    11111  I
                              111111 :     I
                        111111 I     :     I
                 1111111       I     :     I
         11111111              I     :     I
111111111                      I     :     I
===========================================================================
POINT            =    0.000000D+00  S  =   1.6667D-02
DIRECTION        =    1.000000D+00  H  =   1.0000D-01
F0 ( 5.40D-01, 1.85D+00) F1 ( 5.41D-01, 1.86D+00)
I/O:  25  6  6  1  2  3 27 28 29 30 NRML ON   CURRENT  CALLS =    2424

                        FIGURE 25 AN INVISIBLE GRAPH

          AFTER USING THE FLIP COMMAND ONE OBTAINS:

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> PAGE  57
...                            I     :     I                     1111111111
   ...                         I     :     I              1111111
      ....                     I     :     I       1111111
          ...                  I     :     I1111111
             ....              I     : 11111
                 ....          I  11111    I
                     ....    11111   :     I
                        11111  I     :     I
                    1111     .....   :     I
                1111           I  .....    I
             111               I     : ......
         1111                  I     :     I ......
      111                      I     :