*DECK DPOFS
SUBROUTINE DPOFS (A, LDA, N, V, ITASK, IND, WORK)
C***BEGIN PROLOGUE DPOFS
C***PURPOSE Solve a positive definite symmetric system of linear
C equations.
C***LIBRARY SLATEC
C***CATEGORY D2B1B
C***TYPE DOUBLE PRECISION (SPOFS-S, DPOFS-D, CPOFS-C)
C***KEYWORDS HERMITIAN, LINEAR EQUATIONS, POSITIVE DEFINITE, SYMMETRIC
C***AUTHOR Voorhees, E. A., (LANL)
C***DESCRIPTION
C
C Subroutine DPOFS solves a positive definite symmetric
C NxN system of double precision linear equations using
C LINPACK subroutines DPOCO and DPOSL. That is, if A is an
C NxN double precision positive definite symmetric matrix and if
C X and B are double precision N-vectors, then DPOFS 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 R and R-TRANPOSE. These factors are used to
C find the 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 only to solve (ITASK .GT. 1) will be faster for
C the succeeding solutions. In this case, the contents of A,
C LDA, and N must not have been altered by the user following
C factorization (ITASK=1). IND will not be changed by DPOFS
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. Only
C the upper triangle, including the diagonal, of the
C coefficient matrix need be entered and will subse-
C quently be referenced and changed by the routine.
C on return, A contains in its upper triangle an upper
C triangular matrix R such that A = (R-TRANPOSE) * R .
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. N must be greater
C than or equal to 1. (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.
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
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 or
C is not positive definite. A solution
C has not been computed.
C IND=-10 Warning The solution has no apparent significance.
C The solution may be inaccurate or the
C matrix 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, DPOCO, DPOSL, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800514 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 DPOFS
C
INTEGER LDA,N,ITASK,IND,INFO
DOUBLE PRECISION A(LDA,*),V(*),WORK(*),D1MACH
DOUBLE PRECISION RCOND
CHARACTER*8 XERN1, XERN2
C***FIRST EXECUTABLE STATEMENT DPOFS
IF (LDA.LT.N) THEN
IND = -1
WRITE (XERN1, '(I8)') LDA
WRITE (XERN2, '(I8)') N
CALL XERMSG ('SLATEC', 'DPOFS', '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', 'DPOFS', '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', 'DPOFS', 'ITASK = ' // XERN1 //
* ' IS LESS THAN 1', -3, 1)
RETURN
ENDIF
C
IF (ITASK.EQ.1) THEN
C
C FACTOR MATRIX A INTO R
C
CALL DPOCO(A,LDA,N,RCOND,WORK,INFO)
C
C CHECK FOR POSITIVE DEFINITE MATRIX
C
IF (INFO.NE.0) THEN
IND = -4
CALL XERMSG ('SLATEC', 'DPOFS',
* 'SINGULAR OR NOT POSITIVE DEFINITE - 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.EQ.0) THEN
IND = -10
CALL XERMSG ('SLATEC', 'DPOFS',
* 'SOLUTION MAY HAVE NO SIGNIFICANCE', -10, 0)
ENDIF
ENDIF
C
C SOLVE AFTER FACTORING
C
CALL DPOSL(A,LDA,N,V)
RETURN
END