Visible to Intel only — GUID: GUID-6F3C30BC-C2D3-4ECC-B0A7-DD9754A7A62D
Visible to Intel only — GUID: GUID-6F3C30BC-C2D3-4ECC-B0A7-DD9754A7A62D
?pbcon
Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix.
Syntax
call spbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info )
call dpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info )
call cpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info )
call zpbcon( uplo, n, kd, ab, ldab, anorm, rcond, work, rwork, info )
call pbcon( ab, anorm, rcond [,uplo] [,info] )
Include Files
- mkl.fi, lapack.f90
Description
The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix A:
κ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).
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 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)
call ?pbtrf to compute the Cholesky factorization of A.
Input Parameters
n |
INTEGER. The order of the matrix A; n≥ 0. |
kd |
INTEGER. The number of superdiagonals or subdiagonals in the matrix A; kd≥ 0. |
ldab |
INTEGER. The leading dimension of the array ab. (ldab≥kd +1). |
ab, work |
REAL for spbcon DOUBLE PRECISION for dpbcon COMPLEX for cpbcon DOUBLE COMPLEX for zpbcon. Arrays: ab(ldab,*), work(*). The array ab contains the factored matrix A in band form, as returned by ?pbtrf. The second dimension of ab must be at least max(1, n). 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. |
anorm |
REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. The norm of the original matrix A (see Description). |
iwork |
INTEGER. Workspace array, size at least max(1, n). |
rwork |
REAL for cpbcon DOUBLE PRECISION for zpbcon. 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. |
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 pbcon interface are as follows:
ab |
Holds the array A of size (kd+1,n). |
uplo |
Must be 'U' or 'L'. The default value is 'U'. |
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.