nag_ztbtrs (f07vsc) : NAG Library, Mark 23

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 whether

A

is upper or lower triangular.

$uplo = Nag_Upper$: $A$ is upper triangular.
$uplo = Nag_Lower$: $A$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: trans – Nag_TransTypeInput

On entry: indicates the form of the equations.

$trans = Nag_NoTrans$: The equations are of the form $A X = B$ .
$trans = Nag_Trans$: The equations are of the form $A^{T} X = B$ .
$trans = Nag_ConjTrans$: The equations are of the form $A^{H} X = B$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: diag – Nag_DiagTypeInput

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$diag = Nag_NonUnitDiag$: $A$ is a nonunit triangular matrix.
$diag = Nag_UnitDiag$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag = Nag_NonUnitDiag

Nag_UnitDiag

5: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: kd – IntegerInput

On entry:

k_{d}

, the number of superdiagonals of the matrix

A

uplo = Nag_Upper

, or the number of subdiagonals if

uplo = Nag_Lower

Constraint:

kd \geq 0

7: nrhs – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

8: ab[

\dim

] – const ComplexInput

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

\max (1, pdab \times n)

On entry: the

n

n

triangular band matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

A_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [k_{d} + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{d}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{d})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{d})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [k_{d} + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{d}), \dots, i$ .

diag = Nag_UnitDiag

, the diagonal elements of

AB

are assumed to be

1

, and are not referenced.

9: pdab – IntegerInput

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

A

in the array ab.

Constraint:

pdab \geq kd + 1

10: b[

\dim

] – ComplexInput/Output

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

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

right-hand side matrix

B

On exit: the

n

r

solution matrix

X

11: pdb – IntegerInput

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

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

12: 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,

nrhs = 〈value〉

.
Constraint:

nrhs \geq 0

On entry,

pdab = 〈value〉

.
Constraint:

pdab > 0

On entry,

pdb = 〈value〉

.
Constraint:

pdb > 0

NE_INT_2

On entry,

pdab = 〈value〉

and

kd = 〈value〉

.
Constraint:

pdab \geq kd + 1

On entry,

pdb = 〈value〉

and

n = 〈value〉

.
Constraint:

pdb \geq \max (1, n)

On entry,

pdb = 〈value〉

and

nrhs = 〈value〉

.
Constraint:

pdb \geq \max (1, nrhs)

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_SINGULAR

a (〈value〉, 〈value〉)

is exactly zero.

A

is singular and the solution has not been computed.

7 Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

|E| \leq c (k) ε |A|,

c (k)

is a modest linear function of

k

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (k) cond (A, x) ε,   provided c (k) cond (A, x) ε < 1,

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty}

Note that

cond (A, x) \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

;

cond (A, x)

can be much smaller than

cond (A)

and it is also possible for

cond (A^{H})

, which is the same as

cond (A^{T})

, to be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_ztbrfs (f07vvc), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_ztbcon (f07vuc) with

norm = Nag_InfNorm

9 Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} - 1.94 + 4.43 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ - 3.39 + 3.44 i & 4.12 - 4.27 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 1.62 + 3.68 i & - 1.84 + 5.53 i & 0.43 - 2.66 i & 0.00 + 0.00 i \\ 0.00 + 0.00 i & - 2.77 - 1.93 i & 1.74 - 0.04 i & 0.44 + 0.10 i \end{array})

and

B = (\begin{array}{r} - 8.86 - 3.88 i & - 24.09 - 5.27 i \\ - 15.57 - 23.41 i & - 57.97 + 8.14 i \\ - 7.63 + 22.78 i & 19.09 - 29.51 i \\ - 14.74 - 2.40 i & 19.17 + 21.33 i \end{array}) .

Here

A

is treated as a lower triangular band matrix with two subdiagonals.

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_ztbtrs (f07vsc)

+− 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 Documentnag_ztbtrs (f07vsc)

+− 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_ztbtrs (f07vsc)