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

5 Arguments

1: $order$ – 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

order = 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:

order = Nag_RowMajor

Nag_ColMajor

2: $uplo$ – Nag_UploType Input

On entry: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $ap [\dim]$ – const double Input

Note: the dimension, dim, of the array ap must be at least

\max (1, n \times (n + 1) / 2)

On entry: the Cholesky factor of

A

stored in packed form, as returned by f07gdc.

5: $anorm$ – double Input

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling f16rdc with its argument

norm = Nag_OneNorm

. anorm must be computed either before calling f07gdc or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

6: $rcond$ – 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: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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, $anorm = 〈value〉$ .
Constraint: $anorm \geq 0.0$ .

7 Accuracy

The computed estimate rcond is never less than the true value

ρ

, and in practice is nearly always less than

10 ρ

, although examples can be constructed where rcond is much larger.

8 Parallelism and Performance

f07ggc 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.

9 Further Comments

A call to f07ggc involves solving a number of systems of linear equations of the form

A x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n^{2}

floating-point operations but takes considerably longer than a call to f07gec with one right-hand side, because extra care is taken to avoid overflow when

A

is approximately singular.

The complex analogue of this function is f07guc.

10 Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) .

Here

A

is symmetric positive definite, stored in packed form, and must first be factorized by f07gdc. The true condition number in the

1

-norm is

97.32

Interfaces: FL CL AD

NAG CL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07gg: FL CL AD

NAG CL Interfacef07ggc (dppcon)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f07ggc (dppcon)