Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
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

lapack_int LAPACKE_spbsvx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
float*
ab
,
lapack_int
ldab
,
float*
afb
,
lapack_int
ldafb
,
char*
equed
,
float*
s
,
float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_dpbsvx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
double*
ab
,
lapack_int
ldab
,
double*
afb
,
lapack_int
ldafb
,
char*
equed
,
double*
s
,
double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
);
lapack_int LAPACKE_cpbsvx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
lapack_complex_float*
ab
,
lapack_int
ldab
,
lapack_complex_float*
afb
,
lapack_int
ldafb
,
char*
equed
,
float*
s
,
lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_zpbsvx
(
int
matrix_layout
,
char
fact
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
lapack_complex_double*
ab
,
lapack_int
ldab
,
lapack_complex_double*
afb
,
lapack_int
ldafb
,
char*
equed
,
double*
s
,
lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
);
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.
  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 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
The leading dimension of
ab
;
ldab
≥
kd
+1.
ldafb
The leading dimension of
afb
;
ldafb
≥
kd
+1.
ldb
The leading dimension of
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
 had an illegal value.
If
info
 =
i
, and
i
≤
n
, the leading minor of order
i
 (and therefore the matrix
A
 itself) is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed;
rcond
 =0 is returned. If
info
 =
i
, and
i
 =
n
 + 1, then
U
 is nonsingular, 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.