sspsvx - use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices
SUBROUTINE SSPSVX(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO INTEGER N, NRHS, LDB, LDX, INFO INTEGER IPIVOT(*), WORK2(*) REAL RCOND REAL AP(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) SUBROUTINE SSPSVX_64(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO INTEGER*8 N, NRHS, LDB, LDX, INFO INTEGER*8 IPIVOT(*), WORK2(*) REAL RCOND REAL AP(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) F95 INTERFACE SUBROUTINE SPSVX(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO INTEGER :: N, NRHS, LDB, LDX, INFO INTEGER, DIMENSION(:) :: IPIVOT, WORK2 REAL :: RCOND REAL, DIMENSION(:) :: AP, AF, FERR, BERR, WORK REAL, DIMENSION(:,:) :: B, X SUBROUTINE SPSVX_64(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO INTEGER(8) :: N, NRHS, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2 REAL :: RCOND REAL, DIMENSION(:) :: AP, AF, FERR, BERR, WORK REAL, DIMENSION(:,:) :: B, X C INTERFACE #include <sunperf.h> void sspsvx(char fact, char uplo, int n, int nrhs, float *ap, float *af, int *ipivot, float *b, int ldb, float *x, int ldx, float *rcond, float *ferr, float *berr, int *info); void sspsvx_64(char fact, char uplo, long n, long nrhs, float *ap, float *af, long *ipivot, float *b, long ldb, float *x, long ldx, float *rcond, float *ferr, float *berr, long *info);
Oracle Solaris Studio Performance Library sspsvx(3P) NAME sspsvx - use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices SYNOPSIS SUBROUTINE SSPSVX(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO INTEGER N, NRHS, LDB, LDX, INFO INTEGER IPIVOT(*), WORK2(*) REAL RCOND REAL AP(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) SUBROUTINE SSPSVX_64(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO INTEGER*8 N, NRHS, LDB, LDX, INFO INTEGER*8 IPIVOT(*), WORK2(*) REAL RCOND REAL AP(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) F95 INTERFACE SUBROUTINE SPSVX(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO INTEGER :: N, NRHS, LDB, LDX, INFO INTEGER, DIMENSION(:) :: IPIVOT, WORK2 REAL :: RCOND REAL, DIMENSION(:) :: AP, AF, FERR, BERR, WORK REAL, DIMENSION(:,:) :: B, X SUBROUTINE SPSVX_64(FACT, UPLO, N, NRHS, AP, AF, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO INTEGER(8) :: N, NRHS, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2 REAL :: RCOND REAL, DIMENSION(:) :: AP, AF, FERR, BERR, WORK REAL, DIMENSION(:,:) :: B, X C INTERFACE #include <sunperf.h> void sspsvx(char fact, char uplo, int n, int nrhs, float *ap, float *af, int *ipivot, float *b, int ldb, float *x, int ldx, float *rcond, float *ferr, float *berr, int *info); void sspsvx_64(char fact, char uplo, long n, long nrhs, float *ap, float *af, long *ipivot, float *b, long ldb, float *x, long ldx, float *rcond, float *ferr, float *berr, long *info); PURPOSE SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. Error bounds on the solution and a condition estimate are also pro- vided. The following steps are performed: 1. If FACT = 'N', the diagonal pivoting method is used to factor A as A = U * D * U**T, if UPLO = 'U', or A = L * D * L**T, if UPLO = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. 2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. 3. The system of equations is solved for X using the factored form of A. 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. ARGUMENTS FACT (input) Specifies whether or not the factored form of A has been sup- plied on entry. = 'F': On entry, AF and IPIVOT contain the factored form of A. AP, AF and IPIVOT will not be modified. = 'N': The matrix A will be copied to AF and factored. UPLO (input) = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) The number of linear equations, i.e., the order of the matrix A. N >= 0. NRHS (input) The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AP (input) Real array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further details. AF (input or output) Real array, dimension (N*(N+1)/2) If FACT = 'F', then AF is an input argument and on entry contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as com- puted by SSPTRF, stored as a packed triangular matrix in the same storage format as A. If FACT = 'N', then AF is an output argument and on exit con- tains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as a packed triangular matrix in the same storage format as A. IPIVOT (input or output) Integer array, dimension (N) If FACT = 'F', then IPIVOT is an input argument and on entry contains details of the inter- changes and the block structure of D, as determined by SSP- TRF. If IPIVOT(k) > 0, then rows and columns k and IPIVOT(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIVOT(k) = IPIVOT(k-1) < 0, then rows and columns k-1 and -IPIVOT(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIVOT(k) = IPIVOT(k+1) < 0, then rows and columns k+1 and -IPIVOT(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. If FACT = 'N', then IPIVOT is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by SSPTRF. B (input) Real array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B. LDB (input) The leading dimension of the array B. LDB >= max(1,N). X (output) Real array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. LDX (input) The leading dimension of the array X. LDX >= max(1,N). RCOND (output) The estimate of the reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particu- lar, if RCOND = 0), the matrix is singular to working preci- sion. This condition is indicated by a return code of INFO > 0. FERR (output) Real array, dimension (NRHS) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corre- sponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. BERR (output) Real array, dimension (NRHS) The componentwise relative back- ward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). WORK (workspace) Real array, dimension(3*N) WORK2 (workspace) Integer array, dimension(N) INFO (output) = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solu- tion and error bounds could not be computed. RCOND = 0 is returned. = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. FURTHER DETAILS The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U': Two-dimensional storage of the symmetric matrix A: a11 a12 a13 a14 a22 a23 a24 a33 a34 (aij = aji) a44 Packed storage of the upper triangle of A: AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] 7 Nov 2015 sspsvx(3P)