Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?hetrf_rook
Computes the bounded Bunch-Kaufman factorization of a complex Hermitian matrix.
Syntax
call chetrf_rook ( uplo , n , a , lda , ipiv , work , lwork , info )
call zhetrf_rook ( uplo , n , a , lda , ipiv , work , lwork , info )
call hetrf_rook ( a [ , uplo ] [ , ipiv ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
chetrf_rook zhetrf_rook hetrf_rook
The routine computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman diagonal pivoting method:
if uplo='U' , \(A = U*D*U^{H}\)
if uplo='L' , \(A = L*D*L^{H},\)
where \(A\) is the input matrix, \(U\) (or \(L\) ) is a product of permutation and unit upper ( or lower) triangular matrices, and \(D\) is a Hermitian block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates whether the upper or lower triangular part of A is stored:
If uplo = 'U' , the array a stores the upper triangular part of the matrix A .
If uplo = 'L' , the array a stores the lower triangular part of the matrix A .
n
INTEGER . The order of matrix A ; n ≥ 0.
a
COMPLEX for chetrf_rook
COMPLEX*16 for zhetrf_rook .
Array a , size (lda,n) .
The array a contains the upper or the lower triangular part of the matrix A (see uplo ).
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.
lda
INTEGER . The leading dimension of a ; at least max(1, n) .
work
COMPLEX for chetrf_rook
COMPLEX*16 for zhetrf_rook .
Array work(*) .
work(*) is a workspace array of dimension at least max(1, lwork) .
lwork
INTEGER . The size of the work array ( lwork ≥ n ).
The length of work . lwork ≥ 1. For best performance lwork ≥ n * nb , where nb is the block size returned by ilaenv .
If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla .
Output Parameters
a
The block diagonal matrix D and the multipliers used to obtain the factor U or L (see Application Notes for further details).
work(1)
If info = 0 , on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.
ipiv
INTEGER .
Array, size at least max(1, n) . Contains details of the interchanges and the block structure of D .
If uplo = ‘U’ :
If ipiv(k) > 0 , then rows and columns k and ipiv(k) were interchanged and Dk, k is a 1-by-1 diagonal block.
If ipiv(k) < 0 and ipiv(k-1) < 0, then rows and columns k and - ipiv(k) were interchanged and rows and columns k - 1 and - ipiv(k-1) were interchanged, \(D_{k - 1:k,k - 1:k}\) is a 2-by-2 diagonal block.
If uplo = ‘L’ :
If ipiv(k) > 0, then rows and columns k and ipiv(k) were interchanged and \(D_{k,k}\) is a 1-by-1 diagonal block.
If ipiv(k) < 0 and ipiv(k+1) < 0, then rows and columns k and - ipiv(k) were interchanged and rows and columns k + 1 and - ipiv(k+1) were interchanged, \(D_{k:k + 1,k:k + 1}\) is a 2-by-2 diagonal block.
info
INTEGER .
If info = 0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
If info = i , \(D_{ii}\) is exactly 0. The factorization has been completed, but the block diagonal matrix \(D\) is exactly singular, and division by 0 will occur if you use \(D\) for solving a system of linear equations.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine hetrf_rook interface are as follows:
a
holds the matrix A of size ( n , n )
ipiv
holds the vector of length n
uplo
must be ‘U’ or ‘L’ . The default value is ‘U’ .
Application Notes
If uplo = ‘U’, then \(A = U D U^{H}\) , where
\(U\) = \(P(n)\)* \(U(n)\)* … * \(P(k)\)\(U(k)\)* …,
i.e., \(U\) is a product of terms \(P(k)\)* \(U(k)\), where k decreases from n to 1 in steps of 1 or 2, and \(D\) is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks \(D(k)\). \(P(k)\) is a permutation matrix as defined by ipiv(k), and \(U(k)\) is a unit upper triangular matrix, such that if the diagonal block \(D(k)\) is of order s ( s = 1 or 2), then
\[U(k) = \begin{pmatrix} I & v & 0 \\ 0 & I & 0 \\ \underbrace{0}_{k-s} & \underbrace{0}_{s} & \underbrace{I}_{n-k} \end{pmatrix} \begin{matrix} & k-s \\ & s \\ & n-k \\ & \end{matrix}\]
If s = 1, \(D(k)\) overwrites \(A(k, k)\), and v overwrites \(A(1:k-1, k)\).
If s = 2, the upper triangle of \(D(k)\) overwrites \(A(k-1, k-1)\), \(A(k-1, k)\), and \(A(k, k)\), and v overwrites \(A(1:k-2, k-1:k)\).
If uplo = ‘L’, then \(A = L D L^{H}\) , where
\(L\) = \(P\) (1)* \(L\) (1)* … * \(P(k)\)* \(L(k)\)* …,
i.e., \(L\) is a product of terms \(P(k)\)* \(L(k)\), where k increases from 1 to n in steps of 1 or 2, and \(D\) is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks \(D(k)\). \(P(k)\) is a permutation matrix as defined by ipiv(k), and \(L(k)\) is a unit lower triangular matrix, such that if the diagonal block \(D(k)\) is of order s ( s = 1 or 2), then
\[L(k) = \begin{pmatrix} I & 0 & 0 \\ 0 & I & 0 \\ \underbrace{0}_{k-1} & \underbrace{v}_{s} & \underbrace{I}_{n-k-s+1} \end{pmatrix} \begin{matrix} & k-1 \\ & s \\ & n-k+s-1 \\ & \end{matrix}\]
If s = 1, \(D(k)\) overwrites \(A(k, k)\), and v overwrites \(A(k+1:n, k)\).
If s = 2, the lower triangle of \(D(k)\) overwrites \(A(k, k)\), \(A(k+1, k)\), and \(A(k+1, k+1)\), and v overwrites \(A(k+2:n, k:k+1)\).