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NAME
dpbsvx - use 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,
SYNOPSIS
SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB,
LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR,
BERR, WORK, IWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
), BERR( * ), FERR( * ), S( * ), WORK( * ), X(
LDX, * )
#include <sunperf.h>
void dpbsvx(char fact, char uplo, int n, int kd, int nrhs,
double *dab, int ldab, double *afb, int ldafb,
char *equed, double *s, double *db, int ldb, dou-
ble *dx, int ldx, double *drcond, double *ferr,
double *berr, int *info) ;
PURPOSE
DPBSVX 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 provided.
DESCRIPTION
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. 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, steps 4-6 are
skipped.
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.
ARGUMENTS
FACT (input) CHARACTER*1
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, AFB contains the fac-
tored form of A. If EQUED = 'Y', the matrix A has
been equilibrated with scaling factors given by S.
AB and AFB will not be modified. = 'N': The
matrix A will be copied to AFB and factored.
= 'E': The matrix A will be equilibrated if
necessary, then copied to AFB and factored.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of
the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if
UPLO = 'U', or the number of subdiagonals if UPLO
= 'L'. KD >= 0.
NRHS (input) INTEGER
The number of right-hand sides, i.e., the number
of columns of the matrices B and X. NRHS >= 0.
AB (input/output) DOUBLE PRECISION array, dimension
(LDAB,N)
On entry, the upper or lower triangle of the sym-
metric 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 AB as follows: if
UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-
KD)<=i<=j; if UPLO = 'L', AB(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).
LDAB (input) INTEGER
The leading dimension of the array A. LDAB >=
KD+1.
AFB (input or output) DOUBLE PRECISION array, dimen-
sion (LDAFB,N)
If FACT = 'F', then AFB is an input argument and
on entry contains the triangular factor U or L
from the Cholesky factorization A = U**T*U or A =
L*L**T of the band matrix A, in the same storage
format as A (see AB). If EQUED = 'Y', then AFB is
the factored form of the equilibrated matrix A.
If FACT = 'N', then AFB is an output argument and
on exit returns the triangular factor U or L from
the Cholesky factorization A = U**T*U or A =
L*L**T.
If FACT = 'E', then AFB is an output argument and
on exit returns the triangular factor U or L from
the Cholesky factorization 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).
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >=
KD+1.
EQUED (input or output) CHARACTER*1
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, dimen-
sion (N)
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) 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 overwritten by diag(S) * B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >=
max(1,N).
X (output) DOUBLE PRECISION array, dimension
(LDX,NRHS)
If INFO = 0, 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) INTEGER
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output) DOUBLE PRECISION
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, and the solution and
error bounds are not computed.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
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) DOUBLE PRECISION array, dimension (NRHS)
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) DOUBLE PRECISION array, dimension
(3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 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. = N+1: RCOND is less than machine preci-
sion. The factorization has been completed, but
the matrix is singular to working precision, and
the solution and error bounds have not been com-
puted.
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.
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