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# dpbsvx

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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, * )

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 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**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 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) 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.

(LDAFB,N)
AFB     (input or output) DOUBLE PRECISION array, dimension
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/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'; other-
wise, S is an output variable.  If FACT = 'F' and
EQUED = 'Y', each element of S must be positive.

(LDB,NRHS)
B       (input/output) DOUBLE PRECISION array, dimension
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 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) 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 accu-
rate, 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) 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 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 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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