# clahef_rk.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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clahef_rk.f

## SYNOPSIS

### Functions/Subroutines

subroutine clahef_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)
CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

## Function/Subroutine Documentation

### subroutine clahef_rk (character UPLO, integer N, integer NB, integer KB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Purpose:

``` CLAHEF_RK computes a partial factorization of a complex Hermitian
matrix A using the bounded Bunch-Kaufman (rook) diagonal
pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**H U22**H )

A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L',
( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.

CLAHEF_RK is an auxiliary routine called by CHETRF_RK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
```

Parameters:

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular
```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.
```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.
```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.

If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.

On exit, contains:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```

E

```          E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
```

IPIV

```          IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step.

If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,N-KB+1:N);
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,N-KB+1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,N-KB+1:N).
If -IPIV(k-1) = k-1, no interchange occurred.

c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,1:KB).
If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k+1) = k+1, no interchange occurred.

c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.
```

W

```          W is COMPLEX array, dimension (LDW,NB)
```

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).
```

INFO

```          INFO is INTEGER
= 0: successful exit

< 0: If INFO = -k, the k-th argument had an illegal value

> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.

NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

December 2016

Contributors:

```  December 2016,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
```

Definition at line 264 of file clahef_rk.f.

## Author

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