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

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
a
;
lda
max(1,
n
)
.
ldb
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
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.