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