NAG CL Interfacef07guc (zppcon)

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1Purpose

f07guc estimates the condition number of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by f07grc, using packed storage.

2Specification

 #include
 void f07guc (Nag_OrderType order, Nag_UploType uplo, Integer n, const Complex ap[], double anorm, double *rcond, NagError *fail)
The function may be called by the names: f07guc, nag_lapacklin_zppcon or nag_zppcon.

3Description

f07guc estimates the condition number (in the $1$-norm) of a complex Hermitian positive definite matrix $A$:
 $κ1(A)=‖A‖1‖A-1‖1 .$
Since $A$ is Hermitian, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
Because ${\kappa }_{1}\left(A\right)$ is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of ${\kappa }_{1}\left(A\right)$.
The function should be preceded by a call to f16udc to compute ${‖A‖}_{1}$ and a call to f07grc to compute the Cholesky factorization of $A$. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$.

4References

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

5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{uplo}$Nag_UploType Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{ap}\left[\mathit{dim}\right]$const Complex Input
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the Cholesky factor of $A$ stored in packed form, as returned by f07grc.
5: $\mathbf{anorm}$double Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling f16udc with its argument ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$. anorm must be computed either before calling f07grc or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.
6: $\mathbf{rcond}$double * Output
On exit: an estimate of the reciprocal of the condition number of $A$. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, $A$ is singular to working precision.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{anorm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

7Accuracy

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

8Parallelism and Performance

f07guc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

A call to f07guc involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ real floating-point operations but takes considerably longer than a call to f07gsc with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this function is f07ggc.

10Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ) .$
Here $A$ is Hermitian positive definite, stored in packed form, and must first be factorized by f07grc. The true condition number in the $1$-norm is $201.92$.

10.1Program Text

Program Text (f07guce.c)

10.2Program Data

Program Data (f07guce.d)

10.3Program Results

Program Results (f07guce.r)