NAG Library Function Document

nag_zgbcon (f07buc)

void	nag_zgbcon (Nag_OrderType order, Nag_NormType norm, Integer n, Integer kl, Integer ku, const Complex ab[], Integer pdab, const Integer ipiv[], double anorm, double rcond, NagError fail)

3 Description

nag_zgbcon (f07buc) estimates the condition number of a complex band matrix

A

, in either the

1

-norm or the

\infty

-norm:

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

Note that

κ_{\infty} (A) = κ_{1} (A^{H})

Because the condition number is infinite if

A

is singular, the function actually returns an estimate of the reciprocal of the condition number.

The function should be preceded by a call to nag_zgb_norm (f16ubc) to compute

{‖A‖}_{1}

{‖A‖}_{\infty}

, and a call to nag_zgbtrf (f07brc) to compute the

L U

factorization of

A

. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate

{‖A^{- 1}‖}_{1}

{‖A^{- 1}‖}_{\infty}

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_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: norm – Nag_NormTypeInput

On entry: indicates whether

κ_{1} (A)

κ_{\infty} (A)

is estimated.

$norm = Nag_OneNorm$: $κ_{1} (A)$ is estimated.
$norm = Nag_InfNorm$: $κ_{\infty} (A)$ is estimated.

Constraint:

norm = Nag_OneNorm

Nag_InfNorm

3: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: kl – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

5: ku – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

6: ab[ $\dim$ ] – const ComplexInput

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

\max (1, pdab \times n)

On entry: the

L U

factorization of

A

, as returned by nag_zgbtrf (f07brc).

7: pdab – IntegerInput

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 2 \times kl + ku + 1

8: ipiv[ $\dim$ ] – const IntegerInput

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

\max (1, n)

On entry: the pivot indices, as returned by nag_zgbtrf (f07brc).

9: anorm – doubleInput

On entry: if

norm = Nag_OneNorm

, the

1

-norm of the original matrix

A

norm = Nag_InfNorm

, the

\infty

-norm of the original matrix

A

anorm may be computed by calling nag_zgb_norm (f16ubc) with the same value for the argument norm.

anorm must be computed either before calling nag_zgbtrf (f07brc) or else from a copy of the original matrix

A

(see Section 9).

Constraint:

anorm \geq 0.0

10: 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.

11: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $kl = 〈value〉$ .
Constraint: $kl \geq 0$ .; On entry, $ku = 〈value〉$ .
Constraint: $ku \geq 0$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $pdab = 〈value〉$ .
Constraint: $pdab > 0$ .
NE_INT_3: On entry, $pdab = 〈value〉$ , $kl = 〈value〉$ and $ku = 〈value〉$ .
Constraint: $pdab \geq 2 \times kl + ku + 1$ .
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.
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 Further Comments

A call to nag_zgbcon (f07buc) involves solving a number of systems of linear equations of the form

A x = b

A^{H} x = b

; the number is usually

5

and never more than

11

. Each solution involves approximately

8 n (2 k_{l} + k_{u})

real floating point operations (assuming

n ≫ k_{l}

and

n ≫ k_{u}

) but takes considerably longer than a call to nag_zgbtrs (f07bsc) 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 nag_dgbcon (f07bgc).

9 Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{array}{r} - 1.65 + 2.26 i & - 2.05 - 0.85 i & 0.97 - 2.84 i & 0.00 + 0.00 i \\ 0.00 + 6.30 i & - 1.48 - 1.75 i & - 3.99 + 4.01 i & 0.59 - 0.48 i \\ 0.00 + 0.00 i & - 0.77 + 2.83 i & - 1.06 + 1.94 i & 3.33 - 1.04 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 4.48 - 1.09 i & - 0.46 - 1.72 i \end{array}) .

NAG Library Function Documentnag_zgbcon (f07buc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_zgbcon (f07buc)