## Developer Reference

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

# ?gesv

Computes the solution to the system of linear equations with a square coefficient matrix A and multiple right-hand sides.

## Syntax

Include Files
• mkl.h
Description
The routine solves for
X
the system of linear equations
A*X
=
B
, where
A
is an
n
-by-
n
matrix, the columns of matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions.
The
LU
decomposition with partial pivoting and row interchanges is used to factor
A
as
A
=
P*L*U
, where
P
is a permutation matrix,
L
is unit lower triangular, and
U
is upper triangular. The factored form of
A
is then used to solve the system of equations
A*X
=
B
.
The
dsgesv
and
zcgesv
are mixed precision iterative refinement subroutines for exploiting fast single precision hardware. They first attempt to factorize the matrix in single precision (
dsgesv
) or single complex precision (
zcgesv
) and use this factorization within an iterative refinement procedure to produce a solution with double precision (
dsgesv
) / double complex precision (
zcgesv
) 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 performance over double precision 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 iterativerefinement 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
).
n
The number of linear equations, that is, the order of the matrix
A
;
n
0.
nrhs
The number of right-hand sides, that is, the number of columns of the matrix
B
;
nrhs
0.
a
The array
a
(size max(1,
lda
*
n
))
contains the
n
-by-
n
coefficient matrix
A
.
b
The array
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
contains the
n
-by-
nrhs
matrix of right hand side matrix
B
.
lda
The leading dimension of the array
a
;
lda
max(1,
n
)
.
ldb
The leading dimension of the array
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
Overwritten by the factors
L
and
U
from the factorization of
A
=
P*L*U
; the unit diagonal elements of
L
are not stored.
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
and
U
from the factorization
A
=
P*L*U
; the unit diagonal elements of
L
are not stored.
b
Overwritten by the solution matrix
X
for
dgesv
,
sgesv
,
zgesv
,
zgesv
. Unchanged for
dsgesv
and
zcgesv
.
ipiv
Array, size at least
max(1,
n
)
. The pivot indices that define the permutation matrix
P
; row
i
of the matrix was interchanged with row
ipiv
[i-1]
. Corresponds to the single precision factorization (if
info
= 0 and
iter
0) or the double precision factorization (if
info
= 0 and
iter
< 0).
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
sgetrf
for
dsgesv
, or
cgetrf
for
zcgesv
• 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
,
U
i
,
i
(computed in double precision for mixed precision subroutines) is exactly zero. The factorization has been completed, but the factor
U
is exactly singular, so the solution could not be 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 reserverd 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