void	f08nac (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, double a[], Integer pda, double wr[], double wi[], double vl[], Integer pdvl, double vr[], Integer pdvr, NagError *fail)

The function may be called by the names: f08nac, nag_lapackeig_dgeev or nag_dgeev.

3 Description

The right eigenvector

v_{j}

A

satisfies

A v_{j} = λ_{j} v_{j}

where

λ_{j}

is the

j

th eigenvalue of

A

. The left eigenvector

u_{j}

A

satisfies

u_{j}^{H} A = λ_{j} u_{j}^{H}

where

u_{j}^{H}

denotes the conjugate transpose of

u_{j}

The matrix

A

is first reduced to upper Hessenberg form by means of orthogonal similarity transformations, and the

Q R

algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form,

T

, with

1 \times 1

and

2 \times 2

blocks on the main diagonal. The eigenvalues are computed from

T

, the

2 \times 2

blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of

T

are computed and backtransformed to the eigenvectors of

A

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: $jobvl$ – Nag_LeftVecsType Input

On entry: if

jobvl = Nag_NotLeftVecs

, the left eigenvectors of

A

are not computed.

jobvl = Nag_LeftVecs

, the left eigenvectors of

A

are computed.

Constraint:

jobvl = Nag_NotLeftVecs

Nag_LeftVecs

3: $jobvr$ – Nag_RightVecsType Input

On entry: if

jobvr = Nag_NotRightVecs

, the right eigenvectors of

A

are not computed.

jobvr = Nag_RightVecs

, the right eigenvectors of

A

are computed.

Constraint:

jobvr = Nag_NotRightVecs

Nag_RightVecs

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $a [\dim]$ – double Input/Output

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

\max (1, pda \times n)

The

(i, j)

th element of the matrix

A

is stored in

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

On entry: the

n \times n

matrix

A

On exit: a has been overwritten.

6: $pda$ – Integer Input

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

Constraint:

pda \geq \max (1, n)

7: $wr [\dim]$ – double Output

8: $wi [\dim]$ – double Output

Note: the dimension, dim, of the arrays wr and wi must be at least

\max (1, n)

On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.

9: $vl [\dim]$ – double Output

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

$\max (1, pdvl \times n)$ when $jobvl = Nag_LeftVecs$ ;
$1$ otherwise.

where

VL (i, j)

appears in this document, it refers to the array element

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

On exit: if

jobvl = Nag_LeftVecs

, the left eigenvectors

u_{j}

are stored one after another in vl, in the same order as their corresponding eigenvalues. If the

j

th eigenvalue is real, then

u_{j} = VL (i, j)

, for

i = 1, 2, \dots, n

. If the

j

th and

(j + 1)

st eigenvalues form a complex conjugate pair, then

u_{j} = VL (i, j) + i \times VL (i, j + 1)

and

u_{j + 1} = VL (i, j) - i \times VL (i, j + 1)

, for

i = 1, 2, \dots, n

jobvl = Nag_NotLeftVecs

, vl is not referenced.

10: $pdvl$ – Integer Input

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

Constraints:

if $jobvl = Nag_LeftVecs$ , $pdvl \geq \max (1, n)$ ;
otherwise $pdvl \geq 1$ .

11: $vr [\dim]$ – double Output

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

$\max (1, pdvr \times n)$ when $jobvr = Nag_RightVecs$ ;
$1$ otherwise.

where

VR (i, j)

appears in this document, it refers to the array element

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

On exit: if

jobvr = Nag_RightVecs

, the right eigenvectors

v_{j}

are stored one after another in vr, in the same order as their corresponding eigenvalues. If the

j

th eigenvalue is real, then

v_{j} = VR (i, j)

, for

i = 1, 2, \dots, n

. If the

j

th and

(j + 1)

st eigenvalues form a complex conjugate pair, then

v_{j} = VR (i, j) + i \times VR (i, j + 1)

and

v_{j + 1} = VR (i, j) - i \times VR (i, j + 1)

, for

i = 1, 2, \dots, n

jobvr = Nag_NotRightVecs

, vr is not referenced.

12: $pdvr$ – Integer Input

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

Constraints:

if $jobvr = Nag_RightVecs$ , $pdvr \geq \max (1, n)$ ;
otherwise $pdvr \geq 1$ .

13: $fail$ – 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 $Q R$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $⟨ value ⟩$ to n of wr and wi contain eigenvalues which have converged.
NE_ENUM_INT_2: On entry, $jobvl = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvl = Nag_LeftVecs$ , $pdvl \geq \max (1, n)$ ;
otherwise $pdvl \geq 1$ .

On entry, $jobvr = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvr = Nag_RightVecs$ , $pdvr \geq \max (1, n)$ ;
otherwise $pdvr \geq 1$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

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

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

On entry, $pdvr = ⟨ value ⟩$ .
Constraint: $pdvr > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(A + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ A ‖}_{2},

and

ε

is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f08nac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08nac 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

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real.

The total number of floating-point operations is proportional to

n^{3}

The complex analogue of this function is f08nnc.

10 Example

This example finds all the eigenvalues and right eigenvectors of the matrix

A = (\begin{array}{r} 0.35 & 0.45 & - 0.14 & - 0.17 \\ 0.09 & 0.07 & - 0.54 & 0.35 \\ - 0.44 & - 0.33 & - 0.03 & 0.17 \\ 0.25 & - 0.32 & - 0.13 & 0.11 \end{array}) .

f08na: FL CL CPP AD

NAG CL Interfacef08nac (dgeev)

▸▿ 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
f08nac (dgeev)