Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?gtcon
Estimates the reciprocal of the condition number of a tridiagonal matrix.
Syntax
call sgtcon ( norm , n , dl , d , du , du2 , ipiv , anorm , rcond , work , iwork , info )
call dgtcon ( norm , n , dl , d , du , du2 , ipiv , anorm , rcond , work , iwork , info )
call cgtcon ( norm , n , dl , d , du , du2 , ipiv , anorm , rcond , work , info )
call zgtcon ( norm , n , dl , d , du , du2 , ipiv , anorm , rcond , work , info )
call gtcon ( dl , d , du , du2 , ipiv , anorm , rcond [ , norm ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
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
CHARACTER*1 . 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
INTEGER . The order of the matrix A ; n ≥ 0.
dl , d , du , du2
REAL for sgtcon
DOUBLE PRECISION for dgtcon
COMPLEX for cgtcon
DOUBLE COMPLEX for zgtcon .
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
INTEGER .
Array, size ( n ). The array of pivot indices, as returned by ?gttrf (Computes the LU factorization of a tridiagonal matrix.) .
anorm
REAL for single precision flavors.
DOUBLE PRECISION for double precision flavors.
The norm of the original matrix A (see Description ) .
work
REAL for sgtcon
DOUBLE PRECISION for dgtcon
COMPLEX for cgtcon
DOUBLE COMPLEX for zgtcon .
Workspace array, size (2* n ).
iwork
INTEGER . Workspace array, size ( n ). Used for real flavors only.
Output Parameters
rcond
REAL for single precision flavors.
DOUBLE PRECISION for double precision flavors.
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.
info
INTEGER .
If info = 0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine gtcon interface are as follows:
dl
Holds the vector of length ( n -1).
d
Holds the vector of length n .
du
Holds the vector of length ( n -1).
du2
Holds the vector of length ( n -2).
ipiv
Holds the vector of length n .
norm
Must be ‘1’ , ‘O’ , or ‘I’ . The default value is ‘1’ .
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.