cpbsvx(3P) Sun Performance Library cpbsvx(3P)NAMEcpbsvx - use the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations A * X =
B,
SYNOPSIS
SUBROUTINE CPBSVX(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
COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL RCOND
REAL S(*), FERR(*), BERR(*), WORK2(*)
SUBROUTINE CPBSVX_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
COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER*8 N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL RCOND
REAL S(*), FERR(*), BERR(*), WORK2(*)
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
COMPLEX, DIMENSION(:) :: WORK
COMPLEX, DIMENSION(:,:) :: A, AF, B, X
INTEGER :: N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL :: RCOND
REAL, DIMENSION(:) :: S, FERR, BERR, WORK2
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
COMPLEX, DIMENSION(:) :: WORK
COMPLEX, DIMENSION(:,:) :: A, AF, B, X
INTEGER(8) :: N, KD, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL :: RCOND
REAL, DIMENSION(:) :: S, FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void cpbsvx(char fact, char uplo, int n, int kd, int nrhs, complex *a,
int lda, complex *af, int ldaf, char *equed, float *s, com‐
plex *b, int ldb, complex *x, int ldx, float *rcond, float
*ferr, float *berr, int *info);
void cpbsvx_64(char fact, char uplo, long n, long kd, long nrhs, com‐
plex *a, long lda, complex *af, long ldaf, char *equed, float
*s, complex *b, long ldb, complex *x, long ldx, float *rcond,
float *ferr, float *berr, long *info);
PURPOSEcpbsvx uses the Cholesky factorization A = U**H*U or A = L*L**H to com‐
pute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N Hermitian 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**H * U, if UPLO = 'U', or
A = L * L**H, 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 Hermitian 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**H*U or A = L*L**H 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**H*U or A = L*L**H.
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**H*U or A = L*L**H of the equilibrated
matrix A (see the description of A for the form of the equi‐
librated 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
argument. 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 indi‐
cated 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 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)
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(2*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 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 band storage scheme is illustrated by the following example, when N
= 6, KD = 2, and UPLO = 'U':
Two-dimensional storage of the Hermitian 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.
6 Mar 2009 cpbsvx(3P)
