dppsvx - use the Cholesky factorization to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmet- ric positive definite matrix stored in packed format and X and B are N- by-NRHS matrices
SUBROUTINE DPPSVX(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO, EQUED INTEGER N, NRHS, LDB, LDX, INFO INTEGER WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(*), AF(*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) SUBROUTINE DPPSVX_64(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO, EQUED INTEGER*8 N, NRHS, LDB, LDX, INFO INTEGER*8 WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(*), AF(*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) F95 INTERFACE SUBROUTINE PPSVX(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER :: N, NRHS, LDB, LDX, INFO INTEGER, DIMENSION(:) :: WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: A, AF, S, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: B, X SUBROUTINE PPSVX_64(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER(8) :: N, NRHS, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: A, AF, S, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: B, X C INTERFACE #include <sunperf.h> void dppsvx(char fact, char uplo, int n, int nrhs, double *a, double *af, char *equed, double *s, double *b, int ldb, double *x, int ldx, double *rcond, double *ferr, double *berr, int *info); void dppsvx_64(char fact, char uplo, long n, long nrhs, double *a, dou- ble *af, char *equed, double *s, double *b, long ldb, double *x, long ldx, double *rcond, double *ferr, double *berr, long *info);
Oracle Solaris Studio Performance Library dppsvx(3P) NAME dppsvx - use the Cholesky factorization to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmet- ric positive definite matrix stored in packed format and X and B are N- by-NRHS matrices SYNOPSIS SUBROUTINE DPPSVX(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO, EQUED INTEGER N, NRHS, LDB, LDX, INFO INTEGER WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(*), AF(*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) SUBROUTINE DPPSVX_64(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO, EQUED INTEGER*8 N, NRHS, LDB, LDX, INFO INTEGER*8 WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(*), AF(*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) F95 INTERFACE SUBROUTINE PPSVX(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER :: N, NRHS, LDB, LDX, INFO INTEGER, DIMENSION(:) :: WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: A, AF, S, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: B, X SUBROUTINE PPSVX_64(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER(8) :: N, NRHS, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: A, AF, S, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: B, X C INTERFACE #include <sunperf.h> void dppsvx(char fact, char uplo, int n, int nrhs, double *a, double *af, char *equed, double *s, double *b, int ldb, double *x, int ldx, double *rcond, double *ferr, double *berr, int *info); void dppsvx_64(char fact, char uplo, long n, long nrhs, double *a, dou- ble *af, char *equed, double *s, double *b, long ldb, double *x, long ldx, double *rcond, double *ferr, double *berr, long *info); PURPOSE dppsvx uses the Cholesky factorization A = U**T*U or A = L*L**T to com- pute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite 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 = 'E', real scaling factors are computed to equilibrate the system: diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T* U, if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular matrix and L is a lower triangular matrix. 3. If the leading i-by-i principal minor is not positive definite, 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. 4. The system of equations is solved for X using the factored form of A. 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. 6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration. ARGUMENTS FACT (input) Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AF contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified. = 'N': The matrix A will be copied to AF and factored. = 'E': The matrix A will be equilibrated if necessary, then 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. A (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The j-th column of A is stored in the array A as follows: if UPLO = 'U', A(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', A(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. See below for further details. A is not modi- fied if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S). AF (input or output) DOUBLE PRECISION array, dimension (N*(N+1)/2) If FACT = 'F', then AF is an input argument and on entry con- tains the triangular factor U or L from the Cholesky factor- ization A = U'*U or A = L*L', in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix A. If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky fac- torization A = U'*U or A = L*L' of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky fac- torization A = U'*U or A = L*L' of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). EQUED (input or output) Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. S (input or output) DOUBLE PRECISION array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is over- written by diag(S) * B. LDB (input) The leading dimension of the array B. LDB >= max(1,N). X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equili- brated system is inv(diag(S))*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 after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indi- cated by a return code of INFO > 0. FERR (output) DOUBLE PRECISION 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 corresponding to X(j), FERR(j) is an esti- mated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest ele- ment 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) DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any ele- ment of A or B that makes X(j) an exact solution). WORK (workspace) DOUBLE PRECISION 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: the leading minor of order i of A is not positive def- inite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine preci- sion, meaning that the matrix is singular to working preci- sion. Nevertheless, the solution and error bounds are com- puted because there are a number of situations where the com- puted 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 = conjg(aji)) a44 Packed storage of the upper triangle of A: A = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] 7 Nov 2015 dppsvx(3P)