void	f07cgc (Nag_NormType norm, Integer n, const double dl[], const double d[], const double du[], const double du2[], const Integer ipiv[], double anorm, double rcond, NagError fail)

The function may be called by the names: f07cgc, nag_lapacklin_dgtcon or nag_dgtcon.

3 Description

f07cgc should be preceded by a call to f07cdc, which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

where

P

is a permutation matrix,

L

is unit lower triangular with at most one nonzero subdiagonal element in each column, and

U

is an upper triangular band matrix, with two superdiagonals. f07cgc then utilizes the factorization to estimate either

{‖ A^{- 1} ‖}_{1}

{‖ A^{- 1} ‖}_{\infty}

, from which the estimate of the reciprocal of the condition number of

A

1 / κ (A)

is computed as either

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

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

1 / κ (A)

is returned, rather than

κ (A)

, since when

A

is singular

κ (A)

is infinite.

Note that

κ_{\infty} (A) = κ_{1} (A^{T})

4 References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Arguments

1: $norm$ – Nag_NormType Input

On entry: specifies the norm to be used to estimate

κ (A)

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

Constraint:

norm = Nag_OneNorm

Nag_InfNorm

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $dl [\dim]$ – const double Input

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

4: $d [\dim]$ – const double Input

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

\max (1, n)

On entry: must contain the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

5: $du [\dim]$ – const double Input

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

elements of the first superdiagonal of

U

6: $du2 [\dim]$ – const double Input

Note: the dimension, dim, of the array du2 must be at least

\max (1, n - 2)

On entry: must contain the

(n - 2)

elements of the second superdiagonal of

U

7: $ipiv [\dim]$ – const Integer Input

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

\max (1, n)

On entry: must contain the

n

pivot indices that define the permutation matrix

P

. At the

i

th step, row

i

of the matrix was interchanged with row

ipiv [i - 1]

, and

ipiv [i - 1]

must always be either

i

(i + 1)

ipiv [i - 1] = i

indicating that a row interchange was not performed.

8: $anorm$ – double Input

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 as demonstrated in Section 10 for the

1

-norm. The

\infty

-norm may be similarly computed by swapping the dl and du arrays in the code for the

1

-norm.

anorm must be computed either before calling f07cdc or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

9: $rcond$ – double * Output

On exit: contains an estimate of the reciprocal condition number.

10: $fail$ – 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

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

8 Parallelism and Performance

f07cgc 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

The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The total number of floating-point operations required to perform a solve is proportional to

n

The complex analogue of this function is f07cuc.

10 Example

This example estimates the condition number in the

1

-norm of the tridiagonal matrix

A

given by

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) .

f07cg: FL CL CPP AD

NAG CL Interfacef07cgc (dgtcon)

▸▿ 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
f07cgc (dgtcon)