Developer Reference for Intel® oneAPI Math Kernel Library for C
?gtcon
Estimates the reciprocal of the condition number of a tridiagonal matrix.
Syntax
lapack_int LAPACKE_sgtcon ( charnorm , lapack_intn , const float*dl , const float*d , const float*du , const float*du2 , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_dgtcon ( charnorm , lapack_intn , const double*dl , const double*d , const double*du , const double*du2 , const lapack_int*ipiv , doubleanorm , double*rcond );
lapack_int LAPACKE_cgtcon ( charnorm , lapack_intn , const lapack_complex_float*dl , const lapack_complex_float*d , const lapack_complex_float*du , const lapack_complex_float*du2 , const lapack_int*ipiv , floatanorm , float*rcond );
lapack_int LAPACKE_zgtcon ( charnorm , lapack_intn , const lapack_complex_double*dl , const lapack_complex_double*d , const lapack_complex_double*du , const lapack_complex_double*du2 , const lapack_int*ipiv , doubleanorm , double*rcond );
Include Files
mkl.h
Description
sgtcon dgtcon cgtcon zgtcon gtcon
The routine estimates the reciprocal of the condition number of a real or complex tridiagonal matrix A in the 1-norm or infinity-norm:
\(\kappa_{1}(A) = ||A||_{1} ||A^{-1}||_{1}\)
\(\kappa_{\infty}(A) = ||A||_{\infty} ||A^{-1}||_{\infty}\)
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} |a_{ij}| or \(||A||_{\infty} = max_{i}\sum_{j} |a_{ij}|)\)
call ?gttrf (Computes the LU factorization of a tridiagonal matrix.) to compute the LU factorization of A .
Input Parameters
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.
dl , d , du , du2
Arrays: dl(n -1) , d(n) , du(n -1) , du2(n -2) .
The array dl contains the (n - 1) multipliers that define the matrix L from the LU factorization of A as computed by ?gttrf (Computes the LU factorization of a tridiagonal matrix.) .
The array d contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A .
The array du contains the ( n - 1) elements of the first superdiagonal of U .
The array du2 contains the ( n - 2) elements of the second superdiagonal of U .
ipiv
Array, size ( n ). The array of pivot indices, as returned by ?gttrf (Computes the LU factorization of a tridiagonal matrix.) .
anorm
The norm of the original matrix A (see Description ) .
work
Workspace array, size (2* n ).
iwork
Workspace array, size ( n ). Used for real flavors only.
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.