Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

?pbtrs

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

Syntax

lapack_int
LAPACKE_spbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
float
*
ab
,
lapack_int
ldab
,
float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dpbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
double
*
ab
,
lapack_int
ldab
,
double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_cpbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
lapack_complex_float
*
ab
,
lapack_int
ldab
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zpbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
lapack_complex_double
*
ab
,
lapack_int
ldab
,
lapack_complex_double
*
b
,
lapack_int
ldb
);
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
 =
U
T
*U
 for real data,
A
 =
U
H
*U
 for complex data
if
uplo
=
'U'
A
 =
L*L
T
 for real data,
A
 =
L*L
H
 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
The leading dimension of
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
 had an illegal value.
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|U
H
||U| or |E| ≤ c(kd + 1)ε P|L
H
||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:
Equation
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

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