Developer Reference

  • 0.10
  • 10/21/2020
  • Public Content
Contents

?posv

Computes the solution to the system of linear equations with a symmetric or Hermitian positive-definite coefficient matrix A and multiple right-hand sides.

Syntax

lapack_int
LAPACKE_sposv
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
float
*
a
,
lapack_int
lda
,
float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dposv
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
double
*
a
,
lapack_int
lda
,
double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_cposv
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_float
*
a
,
lapack_int
lda
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zposv
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_double
*
a
,
lapack_int
lda
,
lapack_complex_double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dsposv
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
double
*
a
,
lapack_int
lda
,
double
*
b
,
lapack_int
ldb
,
double
*
x
,
lapack_int
ldx
,
lapack_int
*
iter
);
lapack_int
LAPACKE_zcposv
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
lapack_complex_double
*
a
,
lapack_int
lda
,
lapack_complex_double
*
b
,
lapack_int
ldb
,
lapack_complex_double
*
x
,
lapack_int
ldx
,
lapack_int
*
iter
);
Include Files
  • mkl.h
Description
The routine solves for
X
the real or complex system of linear equations
A*X
=
B
, where
A
is an
n
-by-
n
symmetric/Hermitian positive-definite matrix, the columns of matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions.
The Cholesky decomposition is used to factor
A
as
A
=
U
T
*U
(real flavors) and
A
=
U
H
*U
(complex flavors), if
uplo
=
'U'
or
A
=
L*L
T
(real flavors) and
A
=
L*L
H
(complex flavors), if
uplo
=
'L'
,
where
U
is an upper triangular matrix and
L
is a lower triangular matrix. The factored form of
A
is then used to solve the system of equations
A*X
=
B
.
The
dsposv
and
zcposv
are mixed precision iterative refinement subroutines for exploiting fast single precision hardware. They first attempt to factorize the matrix in single precision (
dsposv
) or single complex precision (
zcposv
) and use this factorization within an iterative refinement procedure to produce a solution with double precision (
dsposv
) / double complex precision (
zcposv
) normwise backward error quality (see below). If the approach fails, the method switches to a double precision or double complex precision factorization respectively and computes the solution.
The iterative refinement is not going to be a winning strategy if the ratio single precision/complex performance over double precision/double complex performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to
ilaenv
in the future. At present, iterative refinement is implemented.
The iterative refinement process is stopped if
iter > itermax
or for all the right-hand sides:
rnmr < sqrt(n)*xnrm*anrm*eps*bwdmax
,
where
  • iter
    is the number of the current iteration in the iterative refinement process
  • rnmr
    is the infinity-norm of the residual
  • xnrm
    is the infinity-norm of the solution
  • anrm
    is the infinity-operator-norm of the matrix
    A
  • eps
    is the machine epsilon returned by
    dlamch
    (‘Epsilon’).
The values
itermax
and
bwdmax
are fixed to 30 and 1.0d+00 respectively.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
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.
nrhs
The number of right-hand sides, the number of columns in
B
;
nrhs
0
.
a
,
b
Arrays:
a
(size max(1,
lda
))
,
b
, size max(
ldb
*
nrhs
) for column major layout and max(
ldb
*
n
) for row major layout,
. The array
a
contains the upper or the lower triangular part of the matrix
A
(see
uplo
).
Note that in the case of
zcposv
the imaginary parts of the diagonal elements need not be set and are assumed to be zero.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
lda
The leading dimension of
a
;
lda
max(1,
n
)
.
ldb
The leading dimension of
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ldx
The leading dimension of the array
x
;
ldx
max(1,
n
) for column major layout and
ldx
nrhs
for row major layout
.
Output Parameters
a
If
info
= 0, the upper or lower triangular part of
a
is overwritten by the Cholesky factor
U
or
L
, as specified by
uplo
.
If iterative refinement has been successfully used (
info
= 0 and
iter
0), then
A
is unchanged.
If double precision factorization has been used (
info
= 0 and
iter
< 0), then the array
A
contains the factors
L
or
U
from the Cholesky factorization.
b
Overwritten by the solution matrix
X
.
x
Array, size
max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
. If
info
= 0, contains the
n
-by-
nrhs
solution matrix
X
.
iter
If
iter
< 0
: iterative refinement has failed, double precision factorization has been performed
  • If
    iter
    = -1
    : the routine fell back to full precision for implementation- or machine-specific reason
  • If
    iter
    = -2
    : narrowing the precision induced an overflow, the routine fell back to full precision
  • If
    iter
    = -3
    : failure of
    spotrf
    for
    dsposv
    , or
    cpotrf
    for
    zcposv
  • If
    iter
    = -31
    : stop the iterative refinement after the 30th iteration.
If
iter
> 0
: iterative refinement has been successfully used. Returns the number of iterations.
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
, 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 has not been computed.

Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804