SPBSVX - 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 SPBSVX( 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
REAL RCOND
INTEGER IWORK( * )
REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,
* ), BERR( * ), FERR( * ), S( * ),
WORK( * ), X( LDX, * )
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 posi
tive definite band matrix and X and B are N-by-NRHS matri
ces.
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',
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 premulti
plied 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 nec
essary, 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.
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) REAL 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 over
written by diag(S)*A*diag(S).
LDAB (input) INTEGER
The leading dimension of the array A. LDAB >=
KD+1.
AFB (input or output) REAL array, dimension (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').
= '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) REAL array, dimension (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) REAL 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) REAL 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 equilibrated system
is inv(diag(S))*X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output) REAL
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) REAL 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 corre
sponding to X(j), FERR(j) is an estimated 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 reli
able as the estimate for RCOND, and is almost
always a slight overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each
change in any element of A or B that makes X(j) an
exact solution).
WORK (workspace) REAL 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. RCOND = 0 is returned. = N+1: U is nonsin
gular, but RCOND is less than machine precision,
meaning that the matrix is singular to working
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.
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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