spbsvx - 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 symmetric positive definite band matrix and X and B are N-by-NRHS matrices
SUBROUTINE SPBSVX(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO, EQUED INTEGER N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER WORK2(*) REAL RCOND REAL A(LDA,*), AF(LDAF,*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) SUBROUTINE SPBSVX_64(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER*1 FACT, UPLO, EQUED INTEGER*8 N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER*8 WORK2(*) REAL RCOND REAL A(LDA,*), AF(LDAF,*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*) F95 INTERFACE SUBROUTINE PBSVX(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER :: N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER, DIMENSION(:) :: WORK2 REAL :: RCOND REAL, DIMENSION(:) :: S, FERR, BERR, WORK REAL, DIMENSION(:,:) :: A, AF, B, X SUBROUTINE PBSVX_64(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER(8) :: N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: WORK2 REAL :: RCOND REAL, DIMENSION(:) :: S, FERR, BERR, WORK REAL, DIMENSION(:,:) :: A, AF, B, X C INTERFACE #include <sunperf.h> void spbsvx(char fact, char uplo, int n, int kd, int nrhs, float *a, int lda, float *af, int ldaf, char *equed, float *s, float *b, int ldb, float *x, int ldx, float *rcond, float *ferr, float *berr, int *info); void spbsvx_64(char fact, char uplo, long n, long kd, long nrhs, float *a, long lda, float *af, long ldaf, char *equed, float *s, float *b, long ldb, float *x, long ldx, float *rcond, float *ferr, float *berr, long *info);
Oracle Solaris Studio Performance Library spbsvx(3P)
NAME
spbsvx - 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 symmetric
positive definite band matrix and X and B are N-by-NRHS matrices
SYNOPSIS
SUBROUTINE SPBSVX(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF,
EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2,
INFO)
CHARACTER*1 FACT, UPLO, EQUED
INTEGER N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER WORK2(*)
REAL RCOND
REAL A(LDA,*), AF(LDAF,*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*),
WORK(*)
SUBROUTINE SPBSVX_64(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF,
EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2,
INFO)
CHARACTER*1 FACT, UPLO, EQUED
INTEGER*8 N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER*8 WORK2(*)
REAL RCOND
REAL A(LDA,*), AF(LDAF,*), S(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*),
WORK(*)
F95 INTERFACE
SUBROUTINE PBSVX(FACT, UPLO, N, KD, NRHS, A, LDA, AF, LDAF,
EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
WORK2, INFO)
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
INTEGER :: N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: S, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: A, AF, B, X
SUBROUTINE PBSVX_64(FACT, UPLO, N, KD, NRHS, A, LDA, AF,
LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
WORK, WORK2, INFO)
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
INTEGER(8) :: N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: S, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: A, AF, B, X
C INTERFACE
#include <sunperf.h>
void spbsvx(char fact, char uplo, int n, int kd, int nrhs, float *a,
int lda, float *af, int ldaf, char *equed, float *s, float
*b, int ldb, float *x, int ldx, float *rcond, float *ferr,
float *berr, int *info);
void spbsvx_64(char fact, char uplo, long n, long kd, long nrhs, float
*a, long lda, float *af, long ldaf, char *equed, float *s,
float *b, long ldb, float *x, long ldx, float *rcond, float
*ferr, float *berr, long *info);
PURPOSE
spbsvx uses the Cholesky factorization A=U**T*U or A=L*L**T to compute
the solution to a real system of linear equations A*X=B, where A is an
N-by-N symmetric positive definite band matrix 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 band matrix, and L is a
lower triangular band 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.
KD (input)
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 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)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array, except
if FACT = 'F' and EQUED = 'Y', then A must contain the equi-
librated matrix diag(S)*A*diag(S).
The j-th column of A is stored in the j-th column of the
array A as follows:
if UPLO = 'U', A(KD+1+i-j,j)=A(i,j) for max(1,j-KD)<=i<=j;
if UPLO = 'L', A(1+i-j,j) =A(i,j) for j<=i<=min(N,j+KD).
See below for further details.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
diag(S)*A*diag(S).
LDA (input)
The leading dimension of the array A. LDA >= KD+1.
AF (input or output)
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**T*U or A=L*L**T of the band matrix A, in the
same storage format as A (see A). If EQUED = 'Y', 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**T*U or A=L*L**T.
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**T*U or A=L*L**T of the equilibrated matrix A
(see the description of A for the form of the equilibrated
matrix).
LDAF (input)
The leading dimension of the array AF. LDAF >= KD+1.
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)
The scale factors for A; not accessed if EQUED = 'N'. S is an
input argument if FACT = 'F'; otherwise, S is an output argu-
ment. If FACT = 'F' and EQUED = 'Y', each element of S must
be positive.
B (input/output)
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)
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 indicated
by a return code of INFO > 0.
FERR (output)
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 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)
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)
dimension(3*N)
WORK2 (workspace)
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 pre-
cision, meaning that the matrix is singular to working preci-
sion.
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 band storage scheme is illustrated by the following example, when N
= 6, KD = 2, and UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13
a22 a23 a24
a33 a34 a35
a44 a45 a46
a55 a56
(aij=conjg(aji)) a66
Band storage of the upper triangle of A:
* * a13 a24 a35 a46
* a12 a23 a34 a45 a56
a11 a22 a33 a44 a55 a66
Similarly, if UPLO = 'L' the format of A is as follows:
a11 a22 a33 a44 a55 a66
a21 a32 a43 a54 a65 *
a31 a42 a53 a64 * *
Array elements marked * are not used by the routine.
7 Nov 2015 spbsvx(3P)