Developer Reference for Intel® oneAPI Math Kernel Library for C
?hetrf_rook
Computes the bounded Bunch-Kaufman factorization of a complex Hermitian matrix.
Syntax
lapack_intLAPACKE_chetrf_rook ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_int*ipiv );
lapack_intLAPACKE_zhetrf_rook ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_int*ipiv );
Include Files
mkl.h
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
matrix_layout
Specifies whether matrix storage layout for array b 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:
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
The order of matrix A ; n ≥ 0.
a
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
The leading dimension of a ; at least max(1, n) .
work
lwork
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
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.
Return Values
This function returns a value info .
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.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
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 \(A(k+2:n, k:k+1)\).