*DECK SPOIR
SUBROUTINE SPOIR (A, LDA, N, V, ITASK, IND, WORK)
C***BEGIN PROLOGUE SPOIR
C***PURPOSE Solve a positive definite symmetric system of linear
C equations. Iterative refinement is used to obtain an error
C estimate.
C***LIBRARY SLATEC
C***CATEGORY D2B1B
C***TYPE SINGLE PRECISION (SPOIR-S, CPOIR-C)
C***KEYWORDS HERMITIAN, LINEAR EQUATIONS, POSITIVE DEFINITE, SYMMETRIC
C***AUTHOR Voorhees, E. A., (LANL)
C***DESCRIPTION
C
C Subroutine SPOIR solves a real positive definite symmetric
C NxN system of single precision linear equations using LINPACK
C subroutines SPOFA and SPOSL. One pass of iterative refine-
C ment is used only to obtain an estimate of the accuracy. That
C is, if A is an NxN real positive definite symmetric matrix
C and if X and B are real N-vectors, then SPOIR solves the
C equation
C
C A*X=B.
C
C The matrix A is first factored into upper and lower
C triangular matrices R and R-TRANSPOSE. These
C factors are used to calculate the solution, X.
C Then the residual vector is found and used
C 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 only solve (ITASK .GT. 1) will be faster for
C the succeeding solutions. In this case, the contents of A,
C LDA, N, and WORK must not have been altered by the user
C following factorization (ITASK=1). IND will not be changed
C by SPOIR in this case.
C
C Argument Description ***
C A REAL(LDA,N)
C the doubly subscripted array with dimension (LDA,N)
C which contains the coefficient matrix. Only the
C upper triangle, including the diagonal, of the
C coefficient matrix need be entered. 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. N must be greater than
C or equal to one. (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 terminal error 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
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 or is not positive definite.
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 DSDOT, R1MACH, SASUM, SCOPY, SPOFA, SPOSL, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800528 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 SPOIR
C
INTEGER LDA,N,ITASK,IND,INFO,J
REAL A(LDA,*),V(*),WORK(N,*),SASUM,XNORM,DNORM,R1MACH
DOUBLE PRECISION DSDOT
CHARACTER*8 XERN1, XERN2
C***FIRST EXECUTABLE STATEMENT SPOIR
IF (LDA.LT.N) THEN
IND = -1
WRITE (XERN1, '(I8)') LDA
WRITE (XERN2, '(I8)') N
CALL XERMSG ('SLATEC', 'SPOIR', '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', 'SPOIR', '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', 'SPOIR', '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 R
CALL SPOFA(WORK,N,N,INFO)
C
C CHECK FOR SINGULAR OR NOT POS.DEF. MATRIX
IF (INFO.NE.0) THEN
IND = -4
CALL XERMSG ('SLATEC', 'SPOIR',
* 'SINGULAR OR NOT POSITIVE DEFINITE - NO SOLUTION', -4, 1)
RETURN
ENDIF
ENDIF
C
C SOLVE AFTER FACTORING
C MOVE VECTOR B TO WORK
C
CALL SCOPY(N,V(1),1,WORK(1,N+1),1)
CALL SPOSL(WORK,N,N,V)
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) = -WORK(J,N+1)
1 +DSDOT(J-1,A(1,J),1,V(1),1)
2 +DSDOT(N-J+1,A(J,J),LDA,V(J),1)
40 CONTINUE
C
C SOLVE A*DELTA=R
C
CALL SPOSL(WORK,N,N,WORK(1,N+1))
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', 'SPOIR',
* 'SOLUTION MAY HAVE NO SIGNIFICANCE', -10, 0)
ENDIF
RETURN
END