Developer Reference for Intel® oneAPI Math Kernel Library for C
?hecon
Estimates the reciprocal of the condition number of a Hermitian matrix.
Syntax
lapack_int LAPACKE_checon ( intmatrix_layout , charuplo , lapack_intn , const lapack_complex_float*a , lapack_intlda , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_zhecon ( intmatrix_layout , charuplo , lapack_intn , const lapack_complex_double*a , lapack_intlda , const lapack_int*ipiv , doubleanorm , double*rcond );
Include Files
mkl.h
Description
checon zhecon hecon
The routine estimates the reciprocal of the condition number of a Hermitian matrix A :
κ_{1}(A) = ||A||_{1} ||A^{-1}||_{1} (since A is Hermitian, \(\kappa_{\infty}(A) = \kappa_{1}(A)\) ).
Before calling this routine:
compute anorm (either || A || 1 =max j Σ i | aij | or \(||A||_{\infty} = max_{i}\sum_{j} |a_{ij}|\) )
call ?hetrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix.) to compute the factorization of A .
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 how the input matrix A has been factored:
If uplo = 'U', the array a stores the upper triangular factor U of the factorization A = U*D*U^{H} .
If uplo = 'L', the array a stores the lower triangular factor L of the factorization A = L*D*L^{H} .
n
The order of matrix A ; n ≥ 0.
a , work
The array a of size max(1, lda * n ) contains the factored matrix A , as returned by GUID-D115CA2C-FB54-4605-9E0C-6152237D63D9.xml#GUID-D115CA2C-FB54-4605-9E0C-6152237D63D9 .
lda
The leading dimension of a ; lda≥ max(1, n) .
ipiv
Array, size at least max(1, n) .
The array ipiv , as returned by ?hetrf (Computes the Bunch-Kaufman factorization of a complex Hermitian matrix.) .
anorm
The norm of the original matrix A (see Description ) .
Output Parameters
rcond
An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.
Return Values
This function returns a value info .
If info = 0 , the execution is successful.
If info = -i , parameter i had an illegal value.
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
The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10 r . A call to this routine involves solving a number of systems of linear equations \(A x = b\) ; the number is usually 5 and never more than 11. Each solution requires approximately \(8n^{2}\) floating-point operations.