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NAG Toolbox

NAG Toolbox: nag_lapack_dppcon (f07gg)

Purpose

nag_lapack_dppcon (f07gg) estimates the condition number of a real symmetric positive definite matrix AA, where AA has been factorized by nag_lapack_dpptrf (f07gd), using packed storage.

Syntax

[rcond, info] = f07gg(uplo, n, ap, anorm)
[rcond, info] = nag_lapack_dppcon(uplo, n, ap, anorm)

Description

nag_lapack_dppcon (f07gg) estimates the condition number (in the 11-norm) of a real symmetric positive definite matrix AA:
κ1(A) = A1A11 .
κ1(A)=A1A-11 .
Since AA is symmetric, κ1(A) = κ(A) = AA1κ1(A)=κ(A)=AA-1.
Because κ1(A)κ1(A) is infinite if AA is singular, the function actually returns an estimate of the reciprocal of κ1(A)κ1(A).

References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how AA has been factorized.
uplo = 'U'uplo='U'
A = UTUA=UTU, where UU is upper triangular.
uplo = 'L'uplo='L'
A = LLTA=LLT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
3:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The Cholesky factor of AA stored in packed form, as returned by nag_lapack_dpptrf (f07gd).
4:     anorm – double scalar
The 11-norm of the original matrix AA, which may be computed by calling nag_blas_dlansp (f06rd) with its parameter norm = '1'norm='1'. anorm must be computed either before calling nag_lapack_dpptrf (f07gd) or else from a copy of the original matrix AA.
Constraint: anorm0.0anorm0.0.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

work iwork

Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of AA. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, AA is singular to working precision.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: anorm, 5: rcond, 6: work, 7: iwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed estimate rcond is never less than the true value ρρ, and in practice is nearly always less than 10ρ10ρ, although examples can be constructed where rcond is much larger.

Further Comments

A call to nag_lapack_dppcon (f07gg) involves solving a number of systems of linear equations of the form Ax = bAx=b; the number is usually 44 or 55 and never more than 1111. Each solution involves approximately 2n22n2 floating point operations but takes considerably longer than a call to nag_lapack_dpptrs (f07ge) with one right-hand side, because extra care is taken to avoid overflow when AA is approximately singular.
The complex analogue of this function is nag_lapack_zppcon (f07gu).

Example

function nag_lapack_dppcon_example
uplo = 'L';
n = int64(4);
ap = [2.039607805437114;
     -1.529705854077835;
     0.2745625891934577;
     -0.04902903378454601;
     1.640121946685673;
     -0.2499814119483738;
     0.6737303907389101;
     0.7887488055748053;
     0.6616575633742563;
     0.5346894269298685];
anorm = 10.16;
[rcond, info] = nag_lapack_dppcon(uplo, n, ap, anorm)
 

rcond =

    0.0103


info =

                    0


function f07gg_example
uplo = 'L';
n = int64(4);
ap = [2.039607805437114;
     -1.529705854077835;
     0.2745625891934577;
     -0.04902903378454601;
     1.640121946685673;
     -0.2499814119483738;
     0.6737303907389101;
     0.7887488055748053;
     0.6616575633742563;
     0.5346894269298685];
anorm = 10.16;
[rcond, info] = f07gg(uplo, n, ap, anorm)
 

rcond =

    0.0103


info =

                    0



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