*DECK SNBIR
SUBROUTINE SNBIR (ABE, LDA, N, ML, MU, V, ITASK, IND, WORK, IWORK)
C***BEGIN PROLOGUE SNBIR
C***PURPOSE Solve a general nonsymmetric banded system of linear
C equations. Iterative refinement is used to obtain an error
C estimate.
C***LIBRARY SLATEC
C***CATEGORY D2A2
C***TYPE SINGLE PRECISION (SNBIR-S, CNBIR-C)
C***KEYWORDS BANDED, LINEAR EQUATIONS, NONSYMMETRIC
C***AUTHOR Voorhees, E. A., (LANL)
C***DESCRIPTION
C
C Subroutine SNBIR solves a general nonsymmetric banded NxN
C system of single precision real linear equations using
C SLATEC subroutines SNBFA and SNBSL. These are adaptations
C of the LINPACK subroutines SGBFA and SGBSL, which require
C a different format for storing the matrix elements.
C One pass of iterative refinement is used only to obtain an
C estimate of the accuracy. If A is an NxN real banded
C matrix and if X and B are real N-vectors, then SNBIR
C solves the equation
C
C A*X=B.
C
C A band matrix is a matrix whose nonzero elements are all
C fairly near the main diagonal, specifically A(I,J) = 0
C if I-J is greater than ML or J-I is greater than
C MU . The integers ML and MU are called the lower and upper
C band widths and M = ML+MU+1 is the total band width.
C SNBIR uses less time and storage than the corresponding
C program for general matrices (SGEIR) if 2*ML+MU .LT. N .
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 find the
C solution vector X . Then the residual vector is found and used
C to calculate an estimate of the relative error, IND . IND esti-
C mates the accuracy of the solution only when the input matrix
C and the right hand side are represented exactly in the computer
C and does not take into account 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 only solve (ITASK .GT. 1) will be faster for
C the succeeding solutions. In this case, the contents of A, LDA,
C N, work and IWORK must not have been altered by the user follow-
C ing factorization (ITASK=1). IND will not be changed by SNBIR
C in this case.
C
C
C Band Storage
C
C If A is a band matrix, the following program segment
C will set up the input.
C
C ML = (band width below the diagonal)
C MU = (band width above the diagonal)
C DO 20 I = 1, N
C J1 = MAX(1, I-ML)
C J2 = MIN(N, I+MU)
C DO 10 J = J1, J2
C K = J - I + ML + 1
C ABE(I,K) = A(I,J)
C 10 CONTINUE
C 20 CONTINUE
C
C This uses columns 1 Through ML+MU+1 of ABE .
C
C Example: If the original matrix is
C
C 11 12 13 0 0 0
C 21 22 23 24 0 0
C 0 32 33 34 35 0
C 0 0 43 44 45 46
C 0 0 0 54 55 56
C 0 0 0 0 65 66
C
C then N = 6, ML = 1, MU = 2, LDA .GE. 5 and ABE should contain
C
C * 11 12 13 , * = not used
C 21 22 23 24
C 32 33 34 35
C 43 44 45 46
C 54 55 56 *
C 65 66 * *
C
C
C Argument Description ***
C
C ABE REAL(LDA,MM)
C on entry, contains the matrix in band storage as
C described above. MM must not be less than M =
C ML+MU+1 . The user is cautioned to dimension ABE
C with care since MM is not an argument and cannot
C be checked by SNBIR. The rows of the original
C matrix are stored in the rows of ABE and the
C diagonals of the original matrix are stored in
C columns 1 through ML+MU+1 of ABE . ABE is
C not altered by the program.
C LDA INTEGER
C the leading dimension of array ABE. 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. N must be greater
C than or equal to 1 . (terminal error message IND=-2)
C ML INTEGER
C the number of diagonals below the main diagonal.
C ML must not be less than zero nor greater than or
C equal to N . (terminal error message IND=-5)
C MU INTEGER
C the number of diagonals above the main diagonal.
C MU must not be less than zero nor greater than or
C equal to N . (terminal error message IND=-6)
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 and IWORK.
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*(NC+1))
C a singly subscripted array of dimension at least
C N*(NC+1) where NC = 2*ML+MU+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 1.
C IND=-3 terminal ITASK is less than 1.
C IND=-4 terminal the matrix A is computationally singular.
C A solution has not been computed.
C IND=-5 terminal ML is less than zero or is greater than
C or equal to N .
C IND=-6 terminal MU is less than zero or is greater than
C or equal to N .
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, SNBFA, SNBSL, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800815 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 SNBIR
C
INTEGER LDA,N,ITASK,IND,IWORK(*),INFO,J,K,KK,L,M,ML,MU,NC
REAL ABE(LDA,*),V(*),WORK(N,*),XNORM,DNORM,SDSDOT,SASUM,R1MACH
CHARACTER*8 XERN1, XERN2
C***FIRST EXECUTABLE STATEMENT SNBIR
IF (LDA.LT.N) THEN
IND = -1
WRITE (XERN1, '(I8)') LDA
WRITE (XERN2, '(I8)') N
CALL XERMSG ('SLATEC', 'SNBIR', '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', 'SNBIR', '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', 'SNBIR', 'ITASK = ' // XERN1 //
* ' IS LESS THAN 1', -3, 1)
RETURN
ENDIF
C
IF (ML.LT.0 .OR. ML.GE.N) THEN
IND = -5
WRITE (XERN1, '(I8)') ML
CALL XERMSG ('SLATEC', 'SNBIR',
* 'ML = ' // XERN1 // ' IS OUT OF RANGE', -5, 1)
RETURN
ENDIF
C
IF (MU.LT.0 .OR. MU.GE.N) THEN
IND = -6
WRITE (XERN1, '(I8)') MU
CALL XERMSG ('SLATEC', 'SNBIR',
* 'MU = ' // XERN1 // ' IS OUT OF RANGE', -6, 1)
RETURN
ENDIF
C
NC = 2*ML+MU+1
IF (ITASK.EQ.1) THEN
C
C MOVE MATRIX ABE TO WORK
C
M=ML+MU+1
DO 10 J=1,M
CALL SCOPY(N,ABE(1,J),1,WORK(1,J),1)
10 CONTINUE
C
C FACTOR MATRIX A INTO LU
C
CALL SNBFA(WORK,N,N,ML,MU,IWORK,INFO)
C
C CHECK FOR COMPUTATIONALLY SINGULAR MATRIX
C
IF (INFO.NE.0) THEN
IND = -4
CALL XERMSG ('SLATEC', 'SNBIR',
* '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,NC+1),1)
CALL SNBSL(WORK,N,N,ML,MU,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
K = MAX(1,ML+2-J)
KK = MAX(1,J-ML)
L = MIN(J-1,ML)+MIN(N-J,MU)+1
WORK(J,NC+1) = SDSDOT(L,-WORK(J,NC+1),ABE(J,K),LDA,V(KK),1)
40 CONTINUE
C
C SOLVE A*DELTA=R
C
CALL SNBSL(WORK,N,N,ML,MU,IWORK,WORK(1,NC+1),0)
C
C FORM NORM OF DELTA
C
DNORM = SASUM(N,WORK(1,NC+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', 'SNBIR',
* 'SOLUTION MAY HAVE NO SIGNIFICANCE', -10, 0)
ENDIF
RETURN
END