Contents

# ?potrs

Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix.

## Syntax

Include Files
• mkl.h
Description
The routine solves for
X
the system of linear equations
A*X
=
B
with a symmetric positive-definite or, for complex data, Hermitian positive-definite matrix
A
, given the Cholesky factorization of
A
:
 A = UT*U for real data, A = UH*U for complex data if uplo='U' A = L*LT for real data, A = L*LH for complex data if uplo='L'
where
L
is a lower triangular matrix and
U
is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix
B
.
Before calling this routine, you must call ?potrf to compute the Cholesky factorization of
A
.
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 how the input matrix
A
has been factored:
If
uplo
=
'U'
,
U
is stored, where
A
=
U
T
*
U
for real data,
A
=
U
H
*
U
for complex data.
If
uplo
=
'L'
,
L
is stored, where
A
=
L
*
L
T
for real data,
A
=
L
*
L
H
for complex data.
n
The order of matrix
A
;
n
0.
nrhs
The number of right-hand sides
(
nrhs
0)
.
a
Array
A
of size at least max(1,
lda
*
n
)
The array
a
contains the factor
U
or
L
(see
uplo
) as returned by potrf. .
lda
a
.
lda
max(1,
n
).
b
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations. The size of
b
must be at least max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
ldb
b
.
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout.
Output Parameters
b
Overwritten by the solution matrix
X
.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
Application Notes
If
uplo
=
'U'
, the computed solution for each right-hand side
b
is the exact solution of a perturbed system of equations
(
A
+
E
)
x
=
b
, where
`|E| ≤c(n)ε |UH||U|`
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision.
A similar estimate holds for
uplo
=
'L'
. If
x
0
is the true solution, the computed solution
x
satisfies this error bound:
where
cond(
A
,
x
)
= || |
A
-1
||
A
| |
x
| ||
/ ||
x
||
||
A
-1
||
||
A
||
=
κ
(
A
).
Note that
cond(
A
,
x
)
can be much smaller than
κ
(
A
). The approximate number of floating-point operations for one right-hand side vector
b
is
2
n
2
for real flavors and
8
n
2
for complex flavors.
To estimate the condition number
κ
(
A
)
, call
?pocon
.
To refine the solution and estimate the error, call
?porfs
.

#### 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