*DECK SGEIR
SUBROUTINE SGEIR (A, LDA, N, V, ITASK, IND, WORK, IWORK)
C***BEGIN PROLOGUE SGEIR
C***PURPOSE Solve a general system of linear equations. Iterative
C refinement is used to obtain an error estimate.
C***LIBRARY SLATEC
C***CATEGORY D2A1
C***TYPE SINGLE PRECISION (SGEIR-S, CGEIR-C)
C***KEYWORDS COMPLEX LINEAR EQUATIONS, GENERAL MATRIX,
C GENERAL SYSTEM OF LINEAR EQUATIONS
C***AUTHOR Voorhees, E. A., (LANL)
C***DESCRIPTION
C
C Subroutine SGEIR solves a general NxN system of single
C precision linear equations using LINPACK subroutines SGEFA and
C SGESL. One pass of iterative refinement is used only to obtain
C an estimate of the accuracy. That is, if A is an NxN real
C matrix and if X and B are real N-vectors, then SGEIR solves
C the equation
C
C A*X=B.
C
C The matrix A is first factored into upper and lower tri-
C angular matrices U and L using partial pivoting. These
C factors and the pivoting information are used to calculate
C the solution, X. Then the residual vector is found and
C used to calculate an estimate of the relative error, IND.
C IND estimates the accuracy of the solution only when the
C input matrix and the right hand side are represented
C exactly in the computer and does not take into account
C any errors in the input data.
C
C If the equation A*X=B is to be solved for more than one vector
C B, the factoring of A does not need to be performed again and
C the option to solve only (ITASK .GT. 1) will be faster for
C the succeeding solutions. In this case, the contents of A,
C LDA, N, WORK, and IWORK must not have been altered by the
C user following factorization (ITASK=1). IND will not be
C changed by SGEIR in this case.
C
C Argument Description ***
C
C A REAL(LDA,N)
C the doubly subscripted array with dimension (LDA,N)
C which contains the coefficient matrix. A is not
C altered by the routine.
C LDA INTEGER
C the leading dimension of the array A. LDA must be great-
C er than or equal to N. (terminal error message IND=-1)
C N INTEGER
C the order of the matrix A. The first N elements of
C the array A are the elements of the first column of
C matrix A. N must be greater than or equal to 1.
C (terminal error message IND=-2)
C V REAL(N)
C on entry, the singly subscripted array(vector) of di-
C mension N which contains the right hand side B of a
C system of simultaneous linear equations A*X=B.
C on return, V contains the solution vector, X .
C ITASK INTEGER
C If ITASK=1, the matrix A is factored and then the
C linear equation is solved.
C If ITASK .GT. 1, the equation is solved using the existing
C factored matrix A (stored in WORK).
C If ITASK .LT. 1, then terminal error message IND=-3 is
C printed.
C IND INTEGER
C GT. 0 IND is a rough estimate of the number of digits
C of accuracy in the solution, X. IND=75 means
C that the solution vector X is zero.
C LT. 0 see error message corresponding to IND below.
C WORK REAL(N*(N+1))
C a singly subscripted array of dimension at least N*(N+1).
C IWORK INTEGER(N)
C a singly subscripted array of dimension at least N.
C
C Error Messages Printed ***
C
C IND=-1 terminal N is greater than LDA.
C IND=-2 terminal N is less than one.
C IND=-3 terminal ITASK is less than one.
C IND=-4 terminal The matrix A is computationally singular.
C A solution has not been computed.
C IND=-10 warning The solution has no apparent significance.
C The solution may be inaccurate or the matrix
C A may be poorly scaled.
C
C Note- The above terminal(*fatal*) error messages are
C designed to be handled by XERMSG in which
C LEVEL=1 (recoverable) and IFLAG=2 . LEVEL=0
C for warning error messages from XERMSG. Unless
C the user provides otherwise, an error message
C will be printed followed by an abort.
C
C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
C Stewart, LINPACK Users' Guide, SIAM, 1979.
C***ROUTINES CALLED R1MACH, SASUM, SCOPY, SDSDOT, SGEFA, SGESL, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800430 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 900510 Convert XERRWV calls to XERMSG calls. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE SGEIR
C
INTEGER LDA,N,ITASK,IND,IWORK(*),INFO,J
REAL A(LDA,*),V(*),WORK(N,*),XNORM,DNORM,SDSDOT,SASUM,R1MACH
CHARACTER*8 XERN1, XERN2
C***FIRST EXECUTABLE STATEMENT SGEIR
IF (LDA.LT.N) THEN
IND = -1
WRITE (XERN1, '(I8)') LDA
WRITE (XERN2, '(I8)') N
CALL XERMSG ('SLATEC', 'SGEIR', 'LDA = ' // XERN1 //
* ' IS LESS THAN N = ' // XERN2, -1, 1)
RETURN
ENDIF
C
IF (N.LE.0) THEN
IND = -2
WRITE (XERN1, '(I8)') N
CALL XERMSG ('SLATEC', 'SGEIR', 'N = ' // XERN1 //
* ' IS LESS THAN 1', -2, 1)
RETURN
ENDIF
C
IF (ITASK.LT.1) THEN
IND = -3
WRITE (XERN1, '(I8)') ITASK
CALL XERMSG ('SLATEC', 'SGEIR', 'ITASK = ' // XERN1 //
* ' IS LESS THAN 1', -3, 1)
RETURN
ENDIF
C
IF (ITASK.EQ.1) THEN
C
C MOVE MATRIX A TO WORK
C
DO 10 J=1,N
CALL SCOPY(N,A(1,J),1,WORK(1,J),1)
10 CONTINUE
C
C FACTOR MATRIX A INTO LU
C
CALL SGEFA(WORK,N,N,IWORK,INFO)
C
C CHECK FOR COMPUTATIONALLY SINGULAR MATRIX
C
IF (INFO.NE.0) THEN
IND = -4
CALL XERMSG ('SLATEC', 'SGEIR',
* 'SINGULAR MATRIX A - NO SOLUTION', -4, 1)
RETURN
ENDIF
ENDIF
C
C SOLVE WHEN FACTORING COMPLETE
C MOVE VECTOR B TO WORK
C
CALL SCOPY(N,V(1),1,WORK(1,N+1),1)
CALL SGESL(WORK,N,N,IWORK,V,0)
C
C FORM NORM OF X0
C
XNORM=SASUM(N,V(1),1)
IF (XNORM.EQ.0.0) THEN
IND = 75
RETURN
ENDIF
C
C COMPUTE RESIDUAL
C
DO 40 J=1,N
WORK(J,N+1) = SDSDOT(N,-WORK(J,N+1),A(J,1),LDA,V,1)
40 CONTINUE
C
C SOLVE A*DELTA=R
C
CALL SGESL(WORK,N,N,IWORK,WORK(1,N+1),0)
C
C FORM NORM OF DELTA
C
DNORM = SASUM(N,WORK(1,N+1),1)
C
C COMPUTE IND (ESTIMATE OF NO. OF SIGNIFICANT DIGITS)
C AND CHECK FOR IND GREATER THAN ZERO
C
IND = -LOG10(MAX(R1MACH(4),DNORM/XNORM))
IF (IND.LE.0) THEN
IND = -10
CALL XERMSG ('SLATEC', 'SGEIR',
* 'SOLUTION MAY HAVE NO SIGNIFICANCE', -10, 0)
ENDIF
RETURN
END