Contents

# ?pbtrs

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

## Syntax

Include Files
• mkl.h
Description
The routine solves for real data a system of linear equations
A*X
=
B
with a symmetric positive-definite or, for complex data, Hermitian positive-definite band 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
?pbtrf
to compute the Cholesky factorization of
A
in the band storage form.
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 in
ab
, where
A
=
U
T
*
U
for real matrices and
A
=
U
H
*
U
for complex matrices.
If
uplo
=
'L'
,
L
is stored in
ab
, where
A
=
L
*
L
T
for real matrices and
A
=
L
*
L
H
for complex matrices.
n
The order of matrix
A
;
n
0.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
0.
nrhs
The number of right-hand sides;
nrhs
0.
ab
Array
ab
is of size max (1,
ldab
*
n
).
The array
ab
contains the Cholesky factor, as returned by the factorization routine, in band storage form.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
b
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
The size of
b
is at least max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
ldab
The leading dimension of the array
ab
;
ldab
kd
+1.
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(kd + 1)ε P|UH||U| or |E| ≤ c(kd + 1)ε P|LH||L|`
c
(
k
)
is a modest linear function of
k
, 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 approximate number of floating-point operations for one right-hand side vector is 4
n
*
kd
for real flavors and 16
n
*
kd
for complex flavors.
To estimate the condition number
κ
(
A
)
, call
?pbcon
.
To refine the solution and estimate the error, call
?pbrfs
.

#### Product and Performance Information

1

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