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

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