Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.