Contents

# ?pptrs

Solves a system of linear equations with a packed 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 packed 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
?pptrf
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).
ap
,
b
The size of
ap
must be at least max(1,
n
(
n
+1)/2).
The array
ap
contains the factor
U
or
L
, as specified by
uplo
, in packed storage (see Matrix Storage Schemes).
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 matrix
B
whose columns are the right-hand sides for the systems of equations.
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
?ppcon
.
To refine the solution and estimate the error, call
?pprfs
.

#### Product and Performance Information

1

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Notice revision #20110804