Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?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 , mkl_lapack.f90
Description
spbcon dpbcon cpbcon zpbocon pbcon
The routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix A :
κ_{1}(A) = ||A||_{1} ||A^{-1}||_{1} (since A is symmetric or Hermitian, \(\kappa_{\infty}(A) = \kappa_{1}(A)\) ).
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} = max_{j}Σ_{i} |a_{ij}| or \(||A||_{\infty} = max_{i}\sum_{j} |a_{ij}|\) )
call ?pbtrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.) to compute the Cholesky factorization of A .
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates how the input matrix A has been factored:
If uplo = 'U' , A is factored as A = U^{T}*U for real flavors or A = U^{H}*U for complex flavors, and U is stored.
If uplo = 'L' , A is factored as A = L*L^{T} for real flavors or A = L*L^{H} for complex flavors, and L is stored.
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 GUID-E040EE45-DBF5-4A09-A319-15DA1D1B8F4D.xml#GUID-E040EE45-DBF5-4A09-A319-15DA1D1B8F4D . 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.
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 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 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 4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.