Contents
cpbsvx - 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,
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, complex *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, complex *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);
cpbsvx uses 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, where A is an N-by-N Hermitian positive defin-
ite band matrix and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are
also provided.
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 fac-
tored 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.
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 fac-
tored 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 Her-
mitian 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 equilibrated 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 contains the triangular factor U or L from
the Cholesky factorization 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 factorization 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 factorization A = U**H*U or A =
L*L**H 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'; other-
wise, 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 overwritten 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 equilibrated 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 par-
ticular, 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 solu-
tion 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 estimated
upper bound for the magnitude of the largest ele-
ment 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 element 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 ille-
gal value
> 0: if INFO = i, and i is
<= N: the leading minor of order i of A is not
positive definite, so the factorization could not
be completed, and the solution has not been com-
puted. RCOND = 0 is returned. = N+1: U is non-
singular, but RCOND is less than machine preci-
sion, meaning that the matrix is singular to work-
ing 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.
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.