Developer Reference for Intel® oneAPI Math Kernel Library for C
?hetrf
Computes the Bunch-Kaufman factorization of a complex Hermitian matrix.
Syntax
lapack_intLAPACKE_chetrf ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_int*ipiv );
lapack_intLAPACKE_zhetrf ( intmatrix_layout , charuplo , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_int*ipiv );
Include Files
mkl.h
Description
chetrf zhetrf hetrf
The routine computes the factorization of a complex Hermitian matrix A using the 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, \(U\) and \(L\) are products of permutation and triangular matrices with unit diagonal (upper triangular for \(U\) and lower triangular for \(L\) ), and \(D\) is a Hermitian block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks. \(U\) and \(L\) have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of \(D\) .
This routine supports the Progress Routine feature. See Progress Routine 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 and how A is factored:
If uplo = 'U' , the array a stores the upper triangular part of the matrix \(A\) , and \(A\) is factored as \(U*D*U^{H}\) .
If uplo = 'L' , the array a stores the lower triangular part of the matrix \(A\) , and \(A\) is factored as \(L*D*L^{H}\) .
n
The order of matrix A ; n ≥ 0.
a , work
Array , size (lda,*) max(1, lda * n ) .
The array a contains the upper or the lower triangular part of the matrix A (see uplo ).
lda
The leading dimension of a ; at least max(1, n) .
lwork
The size of the work array ( lwork ≥ n ).
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 .
See Application Notes for the suggested value of lwork .
Output Parameters
a
The upper or lower triangular part of a is overwritten by details of the block-diagonal matrix D and the multipliers used to obtain the factor U (or L ).
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
Array, size at least max(1, n) . Contains details of the interchanges and the block structure of D . If ipiv[i-1] = k >0 , then \(d_{ii}\) is a 1-by-1 block, and the i -th row and column of A was interchanged with the k -th row and column.
If uplo = ‘U’ and ipiv [ i ] = ipiv [ i -1] = - m < 0, then D has a 2-by-2 block in rows/columns i and i+1 , and i -th row and column of A was interchanged with the m -th row and column.
If uplo = ‘L’ and ipiv [ i ] = ipiv [ i -1] = - m < 0, then D has a 2-by-2 block in rows/columns i and i +1, and ( i +1)-th row and column of A was interchanged with the m -th row and column.
Return Values
This function returns a value info .
If info = 0 , the execution is successful.
If info = -i , parameter i had an illegal value.
If info = i , \(d_{ii}\) is 0. The factorization has been completed, but D is exactly singular. 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
This routine is suitable for Hermitian matrices that are not known to be positive-definite. If \(A\) is in fact positive-definite, the routine does not perform interchanges, and no 2-by-2 diagonal blocks occur in \(D\) .
The 2-by-2 unit diagonal blocks and the unit diagonal elements of \(U\) and \(L\) are not stored. The remaining elements of \(U\) and \(L\) are stored in the corresponding columns of the array a , but additional row interchanges are required to recover \(U\) or \(L\) explicitly (which is seldom necessary).
If ipiv [ i -1] = i for all i =1...n , then all off-diagonal elements of \(U\) ( \(L\) ) are stored explicitly in the corresponding elements of the array a .
If uplo = 'U' , the computed factors \(U\) and \(D\) are the exact factors of a perturbed matrix \(A + E\) , where
\[|E| \le c(n)\varepsilon P|U||D||U^{T}|P^{T}\]
c(n) is a modest linear function of n , and \(\varepsilon\) is the machine precision.
A similar estimate holds for the computed \(L\) and \(D\) when uplo = 'L' .
The total number of floating-point operations is approximately \((4/3)n^{3}\) .
After calling this routine, you can call the following routines:
?hetrs (Solves a system of linear equations with a \(UDU^H\)- or \(LDL^H\)-factored Hermitian coefficient matrix.)
to solve \(A X = B\)
?hecon (Estimates the reciprocal of the condition number of a Hermitian matrix.)
to estimate the condition number of \(A\)
?hetri (Computes the inverse of a complex Hermitian matrix using \(U*D*U^H\) or \(L*D*L^H\) Bunch-Kaufman factorization.)
to compute the inverse of \(A\) .