*DECK SNBFS SUBROUTINE SNBFS (ABE, LDA, N, ML, MU, V, ITASK, IND, WORK, IWORK) C***BEGIN PROLOGUE SNBFS C***PURPOSE Solve a general nonsymmetric banded system of linear C equations. C***LIBRARY SLATEC C***CATEGORY D2A2 C***TYPE SINGLE PRECISION (SNBFS-S, DNBFS-D, CNBFS-C) C***KEYWORDS BANDED, LINEAR EQUATIONS, NONSYMMETRIC C***AUTHOR Voorhees, E. A., (LANL) C***DESCRIPTION C C Subroutine SNBFS solves a general nonsymmetric banded NxN C system of single precision real linear equations using C SLATEC subroutines SNBCO and SNBSL. These are adaptations C of the LINPACK subroutines SBGCO and SGBSL, which require C a different format for storing the matrix elements. If C A is an NxN real matrix and if X and B are real C N-vectors, then SNBFS 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 SNBFS uses less time and storage than the corresponding C program for general matrices (SGEFS) 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. An approximate condition number is C calculated to provide a rough estimate of the number of C digits of accuracy in the computed solution. 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, C LDA, N and IWORK must not have been altered by the user follow- C ing factorization (ITASK=1). IND will not be changed by SNBFS 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 Furthermore, ML additional columns are needed in C ABE starting with column ML+MU+2 for elements C generated during the triangularization. The total C number of columns needed in ABE is 2*ML+MU+1 . 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 + , + = used for pivoting 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,NC) C on entry, contains the matrix in band storage as C described above. NC must not be less than C 2*ML+MU+1 . The user is cautioned to specify NC C with care since it is not an argument and cannot C be checked by SNBFS. 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 . C on return, contains an upper triangular matrix U and C the multipliers necessary to construct a matrix L C so that A=L*U. 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. C LT. 0 See error message corresponding to IND below. C WORK REAL(N) C a singly subscripted array of dimension at least N. 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, SNBCO, SNBSL, XERMSG C***REVISION HISTORY (YYMMDD) C 800808 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 SNBFS C INTEGER LDA,N,ITASK,IND,IWORK(*),ML,MU REAL ABE(LDA,*),V(*),WORK(*),R1MACH REAL RCOND CHARACTER*8 XERN1, XERN2 C***FIRST EXECUTABLE STATEMENT SNBFS IF (LDA.LT.N) THEN IND = -1 WRITE (XERN1, '(I8)') LDA WRITE (XERN2, '(I8)') N CALL XERMSG ('SLATEC', 'SNBFS', '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', 'SNBFS', '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', 'SNBFS', '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', 'SNBFS', * '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', 'SNBFS', * 'MU = ' // XERN1 // ' IS OUT OF RANGE', -6, 1) RETURN ENDIF C IF (ITASK.EQ.1) THEN C C FACTOR MATRIX A INTO LU C CALL SNBCO(ABE,LDA,N,ML,MU,IWORK,RCOND,WORK) C C CHECK FOR COMPUTATIONALLY SINGULAR MATRIX C IF (RCOND.EQ.0.0) THEN IND = -4 CALL XERMSG ('SLATEC', 'SNBFS', * 'SINGULAR MATRIX A - NO SOLUTION', -4, 1) RETURN ENDIF C C COMPUTE IND (ESTIMATE OF NO. OF SIGNIFICANT DIGITS) C AND CHECK FOR IND GREATER THAN ZERO C IND = -LOG10(R1MACH(4)/RCOND) IF (IND.LE.0) THEN IND = -10 CALL XERMSG ('SLATEC', 'SNBFS', * 'SOLUTION MAY HAVE NO SIGNIFICANCE', -10, 0) ENDIF ENDIF C C SOLVE AFTER FACTORING C CALL SNBSL(ABE,LDA,N,ML,MU,IWORK,V,0) RETURN END