*DECK DGEFS SUBROUTINE DGEFS (A, LDA, N, V, ITASK, IND, WORK, IWORK) C***BEGIN PROLOGUE DGEFS C***PURPOSE Solve a general system of linear equations. C***LIBRARY SLATEC C***CATEGORY D2A1 C***TYPE DOUBLE PRECISION (SGEFS-S, DGEFS-D, CGEFS-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 DGEFS solves a general NxN system of double C precision linear equations using LINPACK subroutines DGECO C and DGESL. That is, if A is an NxN double precision matrix C and if X and B are double precision N-vectors, then DGEFS C solves 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 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 DGEFS C in this case. C C Argument Description *** C C A DOUBLE PRECISION(LDA,N) C on entry, the doubly subscripted array with dimension C (LDA,N) which contains the coefficient matrix. C on return, an upper triangular matrix U and the C multipliers necessary to construct a matrix L C so that A=L*U. 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 the matrix A. N must be greater than or equal to 1. C (terminal error message IND=-2) C V DOUBLE PRECISION(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 DOUBLE PRECISION(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=-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 D1MACH, DGECO, DGESL, XERMSG C***REVISION HISTORY (YYMMDD) C 800326 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 DGEFS C INTEGER LDA,N,ITASK,IND,IWORK(*) DOUBLE PRECISION A(LDA,*),V(*),WORK(*),D1MACH DOUBLE PRECISION RCOND CHARACTER*8 XERN1, XERN2 C***FIRST EXECUTABLE STATEMENT DGEFS IF (LDA.LT.N) THEN IND = -1 WRITE (XERN1, '(I8)') LDA WRITE (XERN2, '(I8)') N CALL XERMSG ('SLATEC', 'DGEFS', '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', 'DGEFS', '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', 'DGEFS', 'ITASK = ' // XERN1 // * ' IS LESS THAN 1', -3, 1) RETURN ENDIF C IF (ITASK.EQ.1) THEN C C FACTOR MATRIX A INTO LU C CALL DGECO(A,LDA,N,IWORK,RCOND,WORK) C C CHECK FOR COMPUTATIONALLY SINGULAR MATRIX C IF (RCOND.EQ.0.0D0) THEN IND = -4 CALL XERMSG ('SLATEC', 'DGEFS', * '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(D1MACH(4)/RCOND) IF (IND.LE.0) THEN IND=-10 CALL XERMSG ('SLATEC', 'DGEFS', * 'SOLUTION MAY HAVE NO SIGNIFICANCE', -10, 0) ENDIF ENDIF C C SOLVE AFTER FACTORING C CALL DGESL(A,LDA,N,IWORK,V,0) RETURN END