# spoir.f

```      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             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
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