ZPBSVX(l) LAPACK driver routine (version 1.1) ZPBSVX(l)
NAME
ZPBSVX - 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 ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S,
B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ),
X( LDX, * )
PURPOSE
ZPBSVX 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 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**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. 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 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) CHARACTER*1
Specifies whether or not the factored form of the matrix A is sup-
plied on entry, and if not, whether the matrix A should be equili-
brated before it is factored. = 'F': On entry, AFB contains the
factored form of A. If EQUED = 'Y', the matrix A has been equili-
brated 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) COMPLEX*16 array, dimension (LDAB,N)
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 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) COMPLEX*16 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**H*U or A = L*L**H of the band matrix A, in the same storage for-
mat 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**H*U or A = L*L**H.
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**H*U or A = L*L**H 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/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 variable if FACT = 'F';
otherwise, it is an output variable.
S (input/output) DOUBLE PRECISION array, dimension (N)
The scale factors for A; not accessed if EQUED = 'N'. S is an
input variable if FACT = 'F'; otherwise, S is an output variable.
If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
B (input/output) COMPLEX*16 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) COMPLEX*16 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 particular, 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 bounds for each solution vector X(j)
(the j-th column of the solution matrix X). If XTRUE is the true
solution, FERR(j) bounds the magnitude of the largest entry in
(X(j) - XTRUE) divided by the magnitude of the largest entry in
X(j). The quality of the error bound depends on the quality of the
estimate of norm(inv(A)) computed in the code; if the estimate of
norm(inv(A)) is accurate, the error bound is guaranteed.
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 entry of A or B
that makes X(j) an exact solution).
WORK (workspace) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: if INFO = i, the leading minor of order i of A is not posi-
tive definite, so the factorization could not be completed, and the
solution has not been computed. = N+1: RCOND is less than machine
precision. The factorization has been completed, but the matrix is
singular to working precision, and the solution and error bounds
have not been computed.
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.
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