void	f07hgc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const double ab[], Integer pdab, double anorm, double rcond, NagError fail)

The function may be called by the names: f07hgc, nag_lapacklin_dpbcon or nag_dpbcon.

3 Description

f07hgc estimates the condition number (in the

1

-norm) of a real symmetric positive definite band matrix

A

κ_{1} (A) = {‖A‖}_{1} {‖A^{- 1}‖}_{1} .

Since

A

is symmetric,

κ_{1} (A) = κ_{\infty} (A) = {‖A‖}_{\infty} {‖A^{- 1}‖}_{\infty}

Because

κ_{1} (A)

is infinite if

A

is singular, the function actually returns an estimate of the reciprocal of

κ_{1} (A)

The function should be preceded by a call to f16rec to compute

{‖A‖}_{1}

and a call to f07hdc 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}

4 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

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: $kd$ – Integer Input

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

5: $ab [\dim]$ – const double Input

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

\max (1, pdab \times n)

On entry: the Cholesky factor of

A

, as returned by f07hdc.

6: $pdab$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.

Constraint:

pdab \geq kd + 1

7: $anorm$ – double Input

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling f16rec with its argument

norm = Nag_OneNorm

. anorm must be computed either before calling f07hdc or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

8: $rcond$ – 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.

9: $fail$ – 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, $kd = 〈value〉$ .
Constraint: $kd \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pdab = 〈value〉$ .
Constraint: $pdab > 0$ .
NE_INT_2: On entry, $pdab = 〈value〉$ and $kd = 〈value〉$ .
Constraint: $pdab \geq kd + 1$ .
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

f07hgc 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 f07hgc 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

4 n k

floating-point operations (assuming

n ≫ k

) but takes considerably longer than a call to f07hec 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 f07huc.

10 Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by f07hdc. The true condition number in the

1

-norm is

74.15

Interfaces: FL CL AD

NAG CL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07hg: FL CL AD

NAG CL Interfacef07hgc (dpbcon)

▸▿ 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
f07hgc (dpbcon)