Developer Reference for Intel® oneAPI Math Kernel Library for C
?sycon
Estimates the reciprocal of the condition number of a symmetric matrix.
Syntax
lapack_int LAPACKE_ssycon ( intmatrix_layout , charuplo , lapack_intn , const float*a , lapack_intlda , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_dsycon ( intmatrix_layout , charuplo , lapack_intn , const double*a , lapack_intlda , const lapack_int*ipiv , doubleanorm , double*rcond );
lapack_int LAPACKE_csycon ( intmatrix_layout , charuplo , lapack_intn , const lapack_complex_float*a , lapack_intlda , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_zsycon ( intmatrix_layout , charuplo , lapack_intn , const lapack_complex_double*a , lapack_intlda , const lapack_int*ipiv , doubleanorm , double*rcond );
Include Files
mkl.h
Description
ssycon dsycon csycon zsycon sycon
The routine estimates the reciprocal of the condition number of a symmetric matrix A :
κ_{1}(A) = ||A||_{1} ||A^{-1}||_{1} (since A is symmetric, \(\kappa_{\infty}(A) = \kappa_{1}(A)\) ).
An estimate is obtained for ||A^{-1}|| , and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^{-1}||) .
Before calling this routine:
compute anorm (either || A || 1 = max j Σ i | aij | or \(||A||_{\infty} = max_{i}\sum_{j} |a_{ij}|\) )
call ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric 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^{T} .
If uplo = 'L' , the array a stores the lower triangular factor L of the factorization A = L*D*L^{T} .
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 ?sytrf .
lda
The leading dimension of a ; lda≥ max(1, n) .
ipiv
Array, size at least max(1, n) .
The array ipiv , as returned by ?sytrf (Computes the Bunch-Kaufman factorization of a symmetric matrix.) .
anorm
The norm of the original matrix A (see Description ) .
iwork
Workspace array, size at least max(1, n) .
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 4 or 5 and never more than 11. Each solution requires approximately \(2n^{2}\) floating-point operations for real flavors and \(8n^{2}\) for complex flavors.