Contents

# ?pbtrf

Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.

## Syntax

Include Files
• mkl.h
Description
The routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix
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.
This routine supports the Progress Routine feature. See Progress Function for details.
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 whether the upper or lower triangular part of
A
is stored in the array
ab
, and how
A
is factored:
If
uplo
=
'U'
, the upper triangle of
A
is stored.
If
uplo
=
'L'
, the lower triangle of
A
is stored.
n
The order of matrix
A
;
n
0.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
0.
ab
Array, size
max(1,
ldab
*
n
)
. The array
ab
contains either the upper or the lower triangular part of the matrix
A
(as specified by
uplo
) in band storage (see Matrix Storage Schemes).
ldab
The leading dimension of the array
ab
.
(
ldab
kd
+ 1)
Output Parameters
ab
The upper or lower triangular part of
A
(in band storage) is overwritten by the Cholesky factor
U
or
L
, as specified by
uplo
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, parameter
i
If
info
=
i
, the leading minor of order
i
(and therefore the matrix
A
itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix
A
.
Application Notes
If
uplo
=
'U'
, the computed factor
U
is the exact factor of a perturbed matrix
A
+
E
, where
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision
.
A similar estimate holds for
uplo
=
'L'
.
The total number of floating-point operations for real flavors is approximately
n
(
kd
+1)
2
. The number of operations for complex flavors is 4 times greater. All these estimates assume that
kd
is much less than
n
.
After calling this routine, you can call the following routines:
to solve
A*X
=
B
to estimate the condition number of
A
.

#### 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