Developer Reference for Intel® oneAPI Math Kernel Library for C
?pbcon
Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix.
Syntax
lapack_int LAPACKE_spbcon ( intmatrix_layout , charuplo , lapack_intn , lapack_intkd , const float*ab , lapack_intldab , floatanorm , float*rcond );
lapack_int LAPACKE_dpbcon ( intmatrix_layout , charuplo , lapack_intn , lapack_intkd , const double*ab , lapack_intldab , doubleanorm , double*rcond );
lapack_int LAPACKE_cpbcon ( intmatrix_layout , charuplo , lapack_intn , lapack_intkd , const lapack_complex_float*ab , lapack_intldab , floatanorm , float*rcond );
lapack_int LAPACKE_zpbcon ( intmatrix_layout , charuplo , lapack_intn , lapack_intkd , const lapack_complex_double*ab , lapack_intldab , doubleanorm , double*rcond );
Include Files
mkl.h
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
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
uplo
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
The order of the matrix A ; n ≥ 0.
kd
The number of superdiagonals or subdiagonals in the matrix A ; kd ≥ 0.
ldab
The leading dimension of the array ab . ( ldab ≥ kd +1).
ab , work
The array ab of size max(1, ldab * n ) contains the factored matrix A in band form, as returned by GUID-E040EE45-DBF5-4A09-A319-15DA1D1B8F4D.xml#GUID-E040EE45-DBF5-4A09-A319-15DA1D1B8F4D .
anorm
The norm of the original matrix A (see Description ) .
iwork
Workspace array, size at least max(1, n) .
rwork
Workspace array, size at least max(1, n) .
Output Parameters
rcond
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.
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , parameter i had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
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.