Contents

# ?hptrs

Solves a system of linear equations with a UDU- or LDL-factored Hermitian coefficient matrix using packed storage.

## Syntax

Include Files
• mkl.h
Description
The routine solves for
X
the system of linear equations
A*X
=
B
with a Hermitian matrix
A
, given the Bunch-Kaufman factorization of
A
:
if
uplo
=
'U'
,
A
=
U*D*U
H
if
uplo
=
'L'
,
A
=
L*D*L
H
,
where
U
and
L
are upper and lower packed triangular matrices with unit diagonal and
D
is a symmetric block-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix
B
.
You must supply to this routine the arrays
ap
(containing
U
or
L
)and
ipiv
in the form returned by the factorization routine
?hptrf
.
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'
, the array
ap
stores the packed factor
U
of the factorization
A
=
U*D*U
H
. If
uplo
=
'L'
, the array
ap
stores the packed factor
L
of the factorization
A
=
L*D*L
H
.
n
The order of matrix
A
;
n
0.
nrhs
The number of right-hand sides;
nrhs
0.
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
array, as returned by
?hptrf
.
ap
The dimension of array
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
contains the matrix
B
whose columns are the right-hand sides for the system of equations.
The size of
b
is 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
For each right-hand side
b
, the computed solution is the exact solution of a perturbed system of equations
(
A
+
E
)
x
=
b
, where
```

|

E

|

≤

c

(

n

)

ε

P

|

U

||

D

||

U

H

|

P

T

or |

E

|

≤

c

(

n

)

ε

P

|

L

||

D

||

L

H

|

P

T```
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision.
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 total number of floating-point operations for one right-hand side vector is approximately
8
n
2
for complex flavors.
To estimate the condition number
κ
(
A
)
, call
?hpcon
.
To refine the solution and estimate the error, call
?hprfs
.

#### Product and Performance Information

1

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Notice revision #20110804