NAG Library Function Document

nag_dpbcon (f07hgc)

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

3 Description

nag_dpbcon (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 nag_dsb_norm (f16rec) to compute

{‖A‖}_{1}

and a call to nag_dpbtrf (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_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: uplo – Nag_UploTypeInput

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 – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: kd – IntegerInput

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

5: ab[ $\dim$ ] – const doubleInput

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 nag_dpbtrf (f07hdc).

6: 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 kd + 1

7: anorm – doubleInput

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling nag_dsb_norm (f16rec) with its argument

norm = Nag_OneNorm

. anorm must be computed either before calling nag_dpbtrf (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 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, $kd = 〈value〉$ .
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.
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_dpbcon (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 nag_dpbtrs (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 nag_zpbcon (f07huc).

9 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 nag_dpbtrf (f07hdc). The true condition number in the

1

-norm is

74.15

NAG Library Function Documentnag_dpbcon (f07hgc)

+− 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_dpbcon (f07hgc)