Developer Reference for Intel® oneAPI Math Kernel Library for C
?gbcon
Estimates the reciprocal of the condition number of a band matrix in the 1-norm or the infinity-norm.
Syntax
lapack_int LAPACKE_sgbcon ( intmatrix_layout , charnorm , lapack_intn , lapack_intkl , lapack_intku , const float*ab , lapack_intldab , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_dgbcon ( intmatrix_layout , charnorm , lapack_intn , lapack_intkl , lapack_intku , const double*ab , lapack_intldab , const lapack_int*ipiv , doubleanorm , double*rcond );
lapack_int LAPACKE_cgbcon ( intmatrix_layout , charnorm , lapack_intn , lapack_intkl , lapack_intku , const lapack_complex_float*ab , lapack_intldab , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_zgbcon ( intmatrix_layout , charnorm , lapack_intn , lapack_intkl , lapack_intku , const lapack_complex_double*ab , lapack_intldab , const lapack_int*ipiv , doubleanorm , double*rcond );
Include Files
mkl.h
Description
The routine estimates the reciprocal of the condition number of a general band matrix A in the 1-norm or infinity-norm:
\(\kappa_{1}(A) = ||A||_{1} ||A^{-1}||_{1} = \kappa_{\infty}(A^{T}) = \kappa_{\infty}(A^{H})\)
\(\kappa_{\infty}(A) = ||A||_{\infty} ||A^{-1}||_{\infty} = \kappa_{1}(A^{T}) = \kappa_{1}(A^{H})\) .
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 ?gbtrf (Computes the LU factorization of a general m-by-n band matrix.) to compute the LU factorization of A .
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
norm
Must be ‘1’ or ‘O’ or ‘I’ .
If norm = ‘1’ or ‘O’ , then the routine estimates the condition number of matrix A in 1-norm.
If norm = 'I' , then the routine estimates the condition number of matrix A in infinity-norm.
n
The order of the matrix A ; n ≥ 0.
kl
The number of subdiagonals within the band of A ; kl ≥ 0.
ku
The number of superdiagonals within the band of A ; ku ≥ 0.
ldab
The leading dimension of the array ab . ( ldab ≥ 2* kl + ku +1).
ipiv
Array, size at least max(1, n) . The ipiv array, as returned by ?gbtrf (Computes the LU factorization of a general m-by-n band matrix.) .
ab , work
The array ab of size max(1, ldab * n ) contains the factored band matrix A , as returned by ?gbtrf .
anorm
The norm of the original matrix A (see gbcon descriptor ) .
iwork
Workspace array, size at least max(1, n) .
rwork
Workspace array, size at least max(1, 2*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 10r . A call to this routine involves solving a number of systems of linear equations \(A x = b\) or \(A^{H} x = b\) ; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n(ku + 2kl) floating-point operations for real flavors and 8n(ku + 2kl) for complex flavors.