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

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