Contents

# ?pbsvx

Uses the Cholesky factorization to compute the solution to the system of linear equations with a symmetric (Hermitian) positive-definite band coefficient matrix A, and provides error bounds on the solution.

## Syntax

Include Files
• mkl.h
Description
The routine uses the Cholesky factorization
A
=
U
T
*U
(real flavors) /
A
=
U
H
*U
(complex flavors) or
A
=
L*L
T
(real flavors) /
A
=
L*L
H
(complex flavors) to compute the solution to a real or complex system of linear equations
A*X
=
B
, where
A
is a
n
-by-
n
symmetric or Hermitian positive definite band matrix, the columns of matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions.
Error bounds on the solution and a condition estimate are also provided.
The routine
?pbsvx
performs the following steps:
1. If
fact
=
'E'
, real scaling factors
s
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
(real),
A
=
U
H
*U
(complex), if
uplo
=
'U'
,
or
A
=
L*L
T
(real),
A
=
L*L
H
(complex), if
uplo
=
'L'
,
where
U
is an upper triangular band matrix and
L
is a lower triangular band matrix.
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 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.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
fact
Must be
'F'
,
'N'
, or
'E'
.
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.
If
fact
=
'F'
: on entry,
afb
contains the factored 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.
If
fact
=
'N'
, the matrix
A
will be copied to
afb
and factored.
If
fact
=
'E'
, the matrix
A
will be equilibrated if necessary, then copied to
afb
and factored.
uplo
Must be
'U'
or
'L'
.
Indicates whether the upper or lower triangular part of
A
is stored:
If
uplo
=
'U'
, the upper triangle of
A
is stored.
If
uplo
=
'L'
, the lower triangle of
A
is stored.
n
The order of matrix
A
;
n
0.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
0.
nrhs
The number of right-hand sides, the number of columns in
B
;
nrhs
0
.
ab
,
afb
,
b
Arrays:
ab
(size max(1,
ldab
*
n
))
,
afb
(size max(1,
ldafb
*
n
))
,
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
.
The array
ab
contains the upper or lower triangle of the matrix
A
in band storage (see Matrix Storage Schemes).
If
fact
=
'F'
and
equed
=
'Y'
, then
ab
must contain the equilibrated matrix
diag
(
s
)*
A
*
diag
(
s
)
.
The array
afb
is an input argument if
fact
=
'F'
. It contains the triangular factor
U
or
L
from the Cholesky factorization of the band matrix
A
in the same storage format as
A
. If
equed
=
'Y'
, then
afb
is the factored form of the equilibrated matrix
A
.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
ldab
ab
;
ldab
kd
+1.
ldafb
afb
;
ldafb
kd
+1.
ldb
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
equed
Must be
'N'
or
'Y'
.
equed
is an input argument if
fact
=
'F'
. It specifies the form of equilibration that was done:
if
equed
=
'N'
, no equilibration was done (always true if
fact
=
'N'
)
if
equed
=
'Y'
, equilibration was done, that is,
A
has been replaced by
diag
(
s
)*
A
*
diag
(
s
)
.
s
Array, size (
n
). The array
s
contains the scale factors for
A
. This array is an input argument if
fact
=
'F'
only; otherwise it is an output argument.
If
equed
=
'N'
,
s
is not accessed.
If
fact
=
'F'
and
equed
=
'Y'
, each element of
s
must be positive.
ldx
The leading dimension of the output array
x
;
ldx
max(1,
n
) for column major layout and
ldx
nrhs
for row major layout
.
Output Parameters
x
Array, size
max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
.
If
info
= 0 or
info
=
n
+1
, the array
x
contains the 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
.
ab
On exit, if
fact
=
'E'
and
equed
=
'Y'
,
A
is overwritten by
diag
(
s
)*
A
*
diag
(
s
).
afb
If
fact
=
'N'
or
'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
(real routines),
A
=
U
H
*U
or
A
=
L*L
H
(complex routines) of the original matrix
A
(if
fact
=
'N'
), or of the equilibrated matrix
A
(if
fact
=
'E'
). See the description of
ab
for the form of the equilibrated matrix.
b
Overwritten by
diag
(
s
)*
B
, if
equed
=
'Y'
; not changed if
equed
=
'N'
.
s
This array is an output argument if
fact
'F'
. See the description of
s
in
Input Arguments
section.
rcond
An 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.
ferr
Array, size at least
max(1,
nrhs
)
. Contains the estimated forward error bound for each solution vector
x
j
(the
j
-th column of the solution matrix
X
). If
xtrue
is the true solution corresponding to
x
j
,
ferr
[
j
-1]
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 reliable as the estimate for
rcond
, and is almost always a slight overestimate of the true error.
berr
Array, size at least
max(1,
nrhs
)
. Contains the component-wise relative backward error for each solution vector
x
j
, that is, the smallest relative change in any element of
A
or
B
that makes
x
j
an exact solution.
equed
If
fact
'F'
, then
equed
is an output argument. It specifies the form of equilibration that was done (see the description of
equed
in
Input Arguments
section).
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i