Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?tpcon
Estimates the reciprocal of the condition number of a packed triangular matrix.
Syntax
call stpcon ( norm , uplo , diag , n , ap , rcond , work , iwork , info )
call dtpcon ( norm , uplo , diag , n , ap , rcond , work , iwork , info )
call ctpcon ( norm , uplo , diag , n , ap , rcond , work , rwork , info )
call ztpcon ( norm , uplo , diag , n , ap , rcond , work , rwork , info )
call tpcon ( ap , rcond [ , uplo ] [ , diag ] [ , norm ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
stpcon dtpcon ctpcon ztpcon tpcon
The routine estimates the reciprocal of the condition number of a packed triangular matrix A in either the 1-norm or infinity-norm:
\(\kappa_{1}(A) = ||A||_{1} ||A^{-1}||_{1} = \kappa_{\infty}(A^{T}) = \kappa_{\infty}(A^{H})\)
\(\kappa_{\infty}(A) = ||A||_{\infty} ||A^{-1}||_{\infty} = \kappa_{1}(A^{T}) = \kappa_{1}(A^{H})\) .
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.
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ . Indicates whether A is upper or lower triangular:
If uplo = 'U' , the array ap stores the upper triangle of A in packed form.
If uplo = 'L' , the array ap stores the lower triangle of A in packed form.
diag
CHARACTER*1 . Must be ‘N’ or ‘U’ .
If diag = 'N' , then A is not a unit triangular matrix.
If diag = 'U' , then A is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array ap .
n
INTEGER . The order of the matrix A ; n ≥ 0.
ap , work
REAL for stpcon
DOUBLE PRECISION for dtpcon
COMPLEX for ctpcon
DOUBLE COMPLEX for ztpcon .
The array ap(*) contains the packed matrix A . The dimension of ap must be at least max(1, n(n+1)/2).
The array work(*) is a workspace for the routine. The dimension of work must be at least max(1, 3*n) for real flavors and max(1, 2*n) for complex flavors.
iwork
INTEGER . Workspace array, size at least max(1, n) .
rwork
REAL for ctpcon
DOUBLE PRECISION for ztpcon .
Workspace array, size at least max(1, n) .
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 tpcon interface are as follows:
ap
Holds the array A of size (n*(n+1)/2) .
norm
Must be ‘1’ , ‘O’ , or ‘I’ . The default value is ‘1’ .
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
diag
Must be ‘N’ or ‘U’ . The default value is ‘N’ .
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 \(n^{2}\) floating-point operations for real flavors and \(4n^{2}\) operations for complex flavors.