Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • 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

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