are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than f08jec, although more storage is required.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $compz$ – Nag_ComputeEigVecsType Input

On entry: indicates whether the eigenvectors are to be computed.

$compz = Nag_NotEigVecs$: Only the eigenvalues are computed (and the array z is not referenced).
$compz = Nag_OrigEigVecs$: The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz = Nag_TridiagEigVecs$: The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the function).

Constraint:

compz = Nag_NotEigVecs

Nag_OrigEigVecs

Nag_TridiagEigVecs

3: $n$ – Integer Input

On entry:

n

, the order of the symmetric tridiagonal matrix

T

Constraint:

n \geq 0

4: $d [\dim]$ – double Input/Output

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

\max (1, n)

On entry: the diagonal elements of the tridiagonal matrix.

On exit: if

fail . code =

NE_NOERROR, the eigenvalues in ascending order.

5: $e [\dim]$ – double Input/Output

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

\max (1, n - 1)

On entry: the subdiagonal elements of the tridiagonal matrix.

On exit: e is overwritten.

6: $z [\dim]$ – double Input/Output

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

$\max (1, pdz \times n)$ when $compz = Nag_OrigEigVecs$ or $Nag_TridiagEigVecs$ ;
$1$ otherwise.

compz = Nag_OrigEigVecs

then the

(i, j)

th element of the matrix

Q

is stored in

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

On entry: if

compz = Nag_OrigEigVecs

, z must contain the orthogonal matrix

Q

used in the reduction to tridiagonal form.

On exit: if

compz = Nag_OrigEigVecs

, z contains the orthonormal eigenvectors of the original symmetric matrix

A

, and if

compz = Nag_TridiagEigVecs

, z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix

T

compz = Nag_NotEigVecs

, z is not referenced.

7: $pdz$ – Integer Input

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

Constraints:

if $compz = Nag_OrigEigVecs$ or $Nag_TridiagEigVecs$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

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_CONVERGENCE: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns $⟨ value ⟩ / (n + 1)$ through $⟨ value ⟩ mod (n + 1)$ .
NE_ENUM_INT_2: On entry, $compz = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $compz = Nag_OrigEigVecs$ or $Nag_TridiagEigVecs$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
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.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(T + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ T ‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue and

{\tilde{λ}}_{i}

is the corresponding computed value, then

| {\tilde{λ}}_{i} - λ_{i} | \leq c (n) ε {‖ T ‖}_{2},

where

c (n)

is a modestly increasing function of

n

z_{i}

is the corresponding exact eigenvector, and

{\tilde{z}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{z}}_{i}, z_{i})

between them is bounded as follows:

θ ({\tilde{z}}_{i}, z_{i}) \leq \frac{c (n) ε {‖ T ‖}_{2}}{\min_{i \neq j} | λ_{i} - λ_{j} |} .

Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

See Section 4.7 of Anderson et al. (1999) for further details. See also f08flc.

8 Parallelism and Performance

f08jhc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jhc 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

If only eigenvalues are required, the total number of floating-point operations is approximately proportional to

n^{2}

. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as f08jec, but for large matrices f08jhc is usually much faster.

The complex analogue of this function is f08jvc.

10 Example

This example finds all the eigenvalues and eigenvectors of the symmetric band matrix

A = (\begin{array}{r} 4.99 & 0.04 & 0.22 & 0 \\ 0.04 & 1.05 & - 0.79 & 1.04 \\ 0.22 & - 0.79 & - 2.31 & - 1.30 \\ 0 & 1.04 & - 1.30 & - 0.43 \end{array}) .

A

is first reduced to tridiagonal form by a call to f08hec.

f08jh: FL CL CPP AD

NAG CL Interfacef08jhc (dstedc)

▸▿ 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
f08jhc (dstedc)