void	f04mcc (Nag_SolveSystem selct, Integer n, Integer nrhs, const double al[], Integer lal, const double d[], const Integer row[], const double b[], Integer tdb, double x[], Integer tdx, NagError *fail)

The function may be called by the names: f04mcc, nag_linsys_real_posdef_vband_solve or nag_real_cholesky_skyline_solve.

3 Description

The normal use of f04mcc is the solution of the systems

A X = B

, following a call of f01mcc to determine the Cholesky factorization

A = L D L^{T}

of the symmetric positive definite variable-bandwidth matrix

A

However, the function may be used to solve any one of the following systems of linear algebraic equations:

{L D L}^{T} X = B ​ (usual system)

(1)

L D X = B ​ (lower triangular system)

(2)

{D L}^{T} X = B ​ (upper triangular system)

(3)

{L L}^{T} X = B

(4)

L X = B ​ (unit lower triangular system)

(5)

L^{T} X = B ​ (unit upper triangular system)

(6)

L

denotes a unit lower triangular variable-bandwidth matrix of order

n

D

a diagonal matrix of order

n

, and

B

a set of right-hand sides.

The matrix

L

is represented by the elements lying within its envelope, i.e., between the first nonzero of each row and the diagonal (see Section 10 for an example). The width

row [i]

of the

i

th row is the number of elements between the first nonzero element and the element on the diagonal inclusive.

4 References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5 Arguments

1: $selct$ – Nag_SolveSystem Input

On entry: selct must specify the type of system to be solved, as follows:

if $selct = Nag_LDLTX$ : solve ${L D L}^{T X} = B$ ;
if $selct = Nag_LDX$ : solve $L D X = B$ ;
if $selct = Nag_DLTX$ : solve ${D L}^{T} X = B$ ;
if $selct = Nag_LLTX$ : solve ${L L}^{T} X = B$ ;
if $selct = Nag_LX$ : solve $L X = B$ ;
if $selct = Nag_LTX$ : solve $L^{T} X = B$ .

Constraint:

selct = Nag_LDLTX

Nag_LDX

Nag_DLTX

Nag_LLTX

Nag_LX

Nag_LTX

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

L

Constraint:

n \geq 1

3: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 1

4: $al [lal]$ – const double Input

On entry: the elements within the envelope of the lower triangular matrix

L

, taken in row by row order, as returned by f01mcc. The unit diagonal elements of

L

must be stored explicitly.

5: $lal$ – Integer Input

On entry: the dimension of the array al.

Constraint:

lal \geq row [0] + row [1] + \dots + row [n - 1]

6: $d [n]$ – const double Input

On entry: the diagonal elements of the diagonal matrix

D

. d is not referenced if

selct = Nag_LLTX

Nag_LX

Nag_LTX

7: $row [n]$ – const Integer Input

On entry:

row [i]

must contain the width of row

i

L

, i.e., the number of elements between the first (left-most) nonzero element and the element on the diagonal, inclusive.

Constraint:

1 \leq row [i] \leq i + 1

for

i = 0, 1, \dots, n - 1

8: $b [n \times tdb]$ – const double Input

Note: the

(i, j)

th element of the matrix

B

is stored in

b [(i - 1) \times tdb + j - 1]

On entry: the

n \times r

right-hand side matrix

B

. See also Section 9.

9: $tdb$ – Integer Input

On entry: the stride separating matrix column elements in the array b.

Constraint:

tdb \geq nrhs

10: $x [n \times tdx]$ – double Output

Note: the

(i, j)

th element of the matrix

X

is stored in

x [(i - 1) \times tdx + j - 1]

On exit: the

n \times r

solution matrix

X

. See also Section 9.

11: $tdx$ – Integer Input

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq nrhs

12: $fail$ – 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_2_INT_ARG_GT: On entry, $row [i] = ⟨ value ⟩$ while $i = ⟨ value ⟩$ . These arguments must satisfy $row [i] \leq i + 1$ .
NE_2_INT_ARG_LT: On entry, $lal = ⟨ value ⟩$ while $row [0] + \dots + row [n - 1] = ⟨ value ⟩$ . These arguments must satisfy $lal \geq row [0] + \dots + row [n - 1]$ .

On entry, $tdb = ⟨ value ⟩$ while $nrhs = ⟨ value ⟩$ . These arguments must satisfy $tdb \geq nrhs$ .

On entry, $tdx = ⟨ value ⟩$ while $nrhs = ⟨ value ⟩$ . These arguments must satisfy $tdx \geq nrhs$ .
NE_BAD_PARAM: On entry, argument selct had an illegal value.
NE_INT_ARG_LT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $nrhs = ⟨ value ⟩$ .
Constraint: $nrhs \geq 1$ .

On entry, $row [⟨ value ⟩]$ must not be less than 1: $row [⟨ value ⟩] = ⟨ value ⟩$ .
NE_NOT_UNIT_DIAG: The lower triangular matrix $L$ has at least one diagonal element which is not equal to unity. The first non-unit element has been located in the array $al [⟨ value ⟩]$ .
NE_ZERO_DIAG: The diagonal matrix $D$ is singular as it has at least one zero element. The first zero element has been located in the array $d [⟨ value ⟩]$ .

7 Accuracy

The usual backward error analysis of the solution of triangular system applies: each computed solution vector is exact for slightly perturbed matrices

L

and

D

, as appropriate (see pages 25-27 and 54-55 of Wilkinson and Reinsch (1971)).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f04mcc is not threaded in any implementation.

9 Further Comments

The time taken by f04mcc is approximately proportional to

p r

, where

p = row [0] + row [1] + \dots + row [n - 1]

The function may be called with the same actual array supplied for the arguments b and x, in which case the solution matrix will overwrite the right-hand side matrix.

10 Example

To solve the system of equations

A X = B

, where

A = (\begin{matrix} 1 & 2 & 0 & 0 & 5 & 0 \\ 2 & 5 & 3 & 0 & 14 & 0 \\ 0 & 3 & 13 & 0 & 18 & 0 \\ 0 & 0 & 0 & 16 & 8 & 24 \\ 5 & 14 & 18 & 8 & 55 & 17 \\ 0 & 0 & 0 & 24 & 17 & 77 \end{matrix}) and B = (\begin{matrix} 6 & −10 \\ 15 & −21 \\ 11 & −3 \\ 0 & 24 \\ 51 & −39 \\ 46 & 67 \end{matrix}) .

Here

A

is symmetric and positive definite and must first be factorized by f01mcc.

f04mc: FL CL CPP AD PY MB

NAG CL Interfacef04mcc (real_​posdef_​vband_​solve)

▸▿ 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
f04mcc (real_posdef_vband_solve)