Cheng S H and Higham N J (1998) A modified Cholesky algorithm based on a symmetric indefinite factorization SIAM J. Matrix Anal. Appl. 19(4) 1097–1110

5 Arguments

1: $uplo$ – Nag_UploType Input

On entry: specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$uplo = Nag_Upper$: The upper triangular part of $A$ is stored and we compute $P^{T} (A + E) P = U D U^{T}$ .
$uplo = Nag_Lower$: The lower triangular part of $A$ is stored and we compute $P^{T} (A + E) P = L D L^{T}$ .

Constraint:

uplo = Nag_Upper

Nag_Lower

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n > 0

3: $a [\dim]$ – double Input/Output

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

\max (1, pda \times n)

On entry: the

n \times n

symmetric matrix

A

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

uplo = Nag_Upper

, the upper triangular part of

A

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

A

must be stored and the elements of the array above the diagonal are not referenced.

On exit: a is overwritten.

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

See Section 9 for further details.

4: $pda$ – Integer Input

On entry: the stride separating row elements of the matrix

A

in the array a.

Constraint:

pda \geq n

5: $offdiag [n]$ – double Output

On exit: the offdiagonals of the symmetric matrix

D

are returned in

offdiag [0], offdiag [1], \dots, offdiag [n - 2]

, for

uplo = Nag_Lower

and in

offdiag [1], offdiag [2], \dots, offdiag [n - 1]

, for

uplo = Nag_Upper

. See Section 9 for further details.

6: $ipiv [n]$ – Integer Output

On exit: gives the permutation information of the factorization. The entries of ipiv are either positive, indicating a

1 \times 1

pivot block, or pairs of negative entries, indicating a

2 \times 2

pivot block.

$ipiv [i - 1] = k > 0$

The

i

th and

k

th rows and columns of

A

were interchanged and

d_{i i}

is a

1 \times 1

block.

$ipiv [i - 1] = - k < 0$ and $ipiv [i] = - ℓ < 0$

The

i

th and

k

th rows and columns, and the

i + 1

st and

ℓ

th rows and columns, were interchanged and

D

has the

2 \times 2

block:

(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})

If $uplo = Nag_Upper$ , $d_{i + 1, i}$ is stored in $offdiag [i]$ . The interchanges were made in the order $i = n, n - 1, \dots, 2$ .
If $uplo = Nag_Lower$ , $d_{i + 1, i}$ is stored in $offdiag [i - 1]$ . The interchanges were made in the order $i = 1, 2, \dots, n - 1$ .

7: $delta$ – double Input

On entry: the value of

δ

Constraint:

delta \geq 0.0

8: $fail$ – 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_EIGENPROBLEM: An intermediate eigenproblem could not be solved. This should not occur. Please contact NAG with details of your call.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq n$ .
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, $delta = ⟨ value ⟩$ .
Constraint: $delta \geq 0.0$ .

7 Accuracy

uplo = Nag_Lower

, the computed factors

L

and

D

are the exact factors not of

P^{T} (A + E) P

but of

P (A + E + F) P^{T}

, where

\begin{array}{l} {‖ F ‖}_{2} & \leq & c (n) ε {‖ A + E ‖}_{2} \\ \leq & c (n) ε {‖ L ‖}_{2} {‖ D ‖}_{2} {‖ L^{T} ‖}_{2}, \end{array}

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

uplo = Nag_Upper

, a similar statement holds for the computed factors

U

and

D

8 Parallelism and Performance

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

f01mdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f01mdc 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 elements of the main diagonal of

D

overwrite the corresponding elements of the main diagonal of

A

; the

n - 1

elements of the subdiagonal (and superdiagonal, by symmetry) elements of

D

are stored in the array offdiag. If

uplo = Nag_Lower

, then these are stored in

offdiag [0], \dots, offdiag [n - 2]

that is

d_{i + 1, i}

, for

i = 1, \dots, n - 1

is stored in

offdiag [i - 1]

; otherwise, they are stored in

offdiag [1], \dots, offdiag [n - 1]

, with

d_{i + 1, i}

stored in

offdiag (i + 1) - 1

The unit diagonal elements of

U

L

are not stored. The remaining elements of

U

L

are stored explicitly in either the strictly upper or strictly lower triangular part of the array a, respectively.

The total number of floating-point operations is approximately

\frac{1}{3} n^{3}

. The searching overhead for rook pivoting used by the algorithm is between

O (n^{2})

and

O (n^{3})

comparisons. Experimental evidence suggests

O (n^{2})

comparisons are usual, see Ashcraft et al. (1998).

All of the entries of the triangular matrix

L

U

are bounded above (by approximately

2.78

), and, therefore, the norm of the matrix itself is also bounded.

The exact size of the perturbation matrix

E

cannot be predicted a priori. However, the algorithm attempts to ensure that it is not much greater than the minimum perturbation

Δ A

such that

A + Δ A

has the minimum eigenvalue

δ

. In particular, it should be zero when

A

is positive definite and

δ = 0

. If

uplo = Nag_Lower

, then in general it can be shown that

{‖ E ‖}_{2} \leq λ_{max} (L L^{T}) (δ - \frac{λ_{min} (A)}{λ_{min} (L L^{T})}),

where

λ_{max}

and

λ_{min}

denote the largest and smallest eigenvalues of the matrix in question. A similar result holds if

uplo = Nag_Upper

10 Example

This example computes the modified Cholesky factorization

A + E = P L D L^{T} P^{T}

, for the indefinite matrix

A

, where

A = (\begin{array}{r} 0.9649 & 0.1419 & 0.0357 & 0.3922 & 0.0462 \\ 0.1419 & 0.4218 & 0.8491 & 0.6555 & 0.0971 \\ 0.0357 & 0.8491 & 0.9340 & 0.1712 & 0.8235 \\ 0.3922 & 0.6555 & 0.1712 & 0.7060 & 0.6948 \\ 0.0462 & 0.0971 & 0.8235 & 0.6948 & 0.3171 \end{array}) .

The output is then passed to f01mec to explicitly form the matrix

A + E

and the norm of

E

is computed.

f01md: FL CL CPP AD PY MB

NAG CL Interfacef01mdc (real_​modified_​cholesky)

▸▿ 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
f01mdc (real_modified_cholesky)