Visible to Intel only — GUID: GUID-A5E155AF-E997-46CD-95BE-F44EDA90B4CA
Visible to Intel only — GUID: GUID-A5E155AF-E997-46CD-95BE-F44EDA90B4CA
?ptcon
Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite tridiagonal matrix.
Syntax
call sptcon( n, d, e, anorm, rcond, work, info )
call dptcon( n, d, e, anorm, rcond, work, info )
call cptcon( n, d, e, anorm, rcond, work, info )
call zptcon( n, d, e, anorm, rcond, work, info )
call ptcon( d, e, anorm, rcond [,info] )
Include Files
- mkl.fi, lapack.f90
Description
The routine computes the reciprocal of the condition number (in the 1-norm) of a real symmetric or complex Hermitian positive-definite tridiagonal matrix using the factorization A = L*D*LT for real flavors and A = L*D*LH for complex flavors or A = UT*D*U for real flavors and A = UH*D*U for complex flavors computed by ?pttrf :
κ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).
The norm ||A-1|| is computed by a direct method, and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A-1||).
Before calling this routine:
compute anorm as ||A||1 = maxjΣi |aij|
call ?pttrf to compute the factorization of A.
Input Parameters
n |
INTEGER. The order of the matrix A; n≥ 0. |
d, work |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Arrays, dimension (n). The array d contains the n diagonal elements of the diagonal matrix D from the factorization of A, as computed by ?pttrf ; work is a workspace array. |
e |
REAL for sptcon DOUBLE PRECISION for dptcon COMPLEX for cptcon DOUBLE COMPLEX for zptcon. Array, size (n -1). Contains off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by ?pttrf . |
anorm |
REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. The 1- norm of the original matrix A (see Description). |
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. |
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:
d |
Holds the vector of length n. |
e |
Holds the vector of length (n-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 10r. 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 4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.