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NAME
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,
SYNOPSIS
SUBROUTINE CPBSVX( 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
REAL RCOND
REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
), WORK( * ), X( LDX, * )
PURPOSE
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.
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 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 neces-
sary, 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 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermi-
tian 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 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 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**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) REAL array, dimension (N)
The scale factors for A; not accessed if EQUED =
'N'. S is an input variable if FACT = 'F'; other-
wise, S is an output variable. If FACT = 'F' and
EQUED = 'Y', each element of S must be positive.
B (input/output) COMPLEX 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 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) 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 partic-
ular, if RCOND = 0), the matrix is singular to work-
ing precision. This condition is indicated by a
return code of INFO > 0, and the solution and error
bounds are not computed.
FERR (output) REAL 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 accu-
rate, the error bound is guaranteed.
BERR (output) REAL 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 array, dimension (2*N)
RWORK (workspace) REAL 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 positive definite, so the factorization could
not be completed, and the solution has not been com-
puted. = 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.