Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

2 Specification

#include <nag.h>

void

f08nbc (Nag_OrderType order, Nag_BalanceType balanc, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_RCondType sense, Integer n, double a[], Integer pda, double wr[], double wi[], double vl[], Integer pdvl, double vr[], Integer pdvr, Integer *ilo, Integer *ihi, double scale[], double *abnrm, double rconde[], double rcondv[], NagError *fail)

The function may be called by the names: f08nbc, nag_lapackeig_dgeevx or nag_dgeevx.

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}

Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation

D A D^{- 1}

, where

D

is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of Anderson et al. (1999).

Following the optional balancing, the matrix

A

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

Q R

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

T

, from which the eigenvalues are computed. Optionally, the eigenvectors of

T

are also computed and backtransformed to those 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: $balanc$ – Nag_BalanceType Input

On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.

$balanc = Nag_NoBalancing$: Do not diagonally scale or permute.
$balanc = Nag_BalancePermute$: Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
$balanc = Nag_BalanceScale$: Diagonally scale the matrix, i.e., replace $A \times D A D^{- 1}$ , where $D$ is a diagonal matrix chosen to make the rows and columns of $A$ more equal in norm. Do not permute.
$balanc = Nag_BalanceBoth$: Both diagonally scale and permute $A$ .

Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

Constraint:

balanc = Nag_NoBalancing

Nag_BalancePermute

Nag_BalanceScale

Nag_BalanceBoth

3: $jobvl$ – 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.

sense = Nag_RCondEigVals

Nag_RCondBoth

, jobvl must be set to

jobvl = Nag_LeftVecs

Constraint:

jobvl = Nag_NotLeftVecs

Nag_LeftVecs

4: $jobvr$ – 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.

sense = Nag_RCondEigVals

Nag_RCondBoth

, jobvr must be set to

jobvr = Nag_RightVecs

Constraint:

jobvr = Nag_NotRightVecs

Nag_RightVecs

5: $sense$ – Nag_RCondType Input

On entry: determines which reciprocal condition numbers are computed.

$sense = Nag_NotRCond$: None are computed.
$sense = Nag_RCondEigVals$: Computed for eigenvalues only.
$sense = Nag_RCondEigVecs$: Computed for right eigenvectors only.
$sense = Nag_RCondBoth$: Computed for eigenvalues and right eigenvectors.

sense = Nag_RCondEigVals

Nag_RCondBoth

, both left and right eigenvectors must also be computed (

jobvl = Nag_LeftVecs

and

jobvr = Nag_RightVecs

Constraint:

sense = Nag_NotRCond

Nag_RCondEigVals

Nag_RCondEigVecs

Nag_RCondBoth

6: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

7: $a [\dim]$ – 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. If

jobvl = Nag_LeftVecs

jobvr = Nag_RightVecs

A

contains the real Schur form of the balanced version of the input matrix

A

8: $pda$ – 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)

9: $wr [\dim]$ – double Output

10: $wi [\dim]$ – 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.

11: $vl [\dim]$ – 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.

12: $pdvl$ – 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 $pdvl \geq 1$ .

13: $vr [\dim]$ – 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.

14: $pdvr$ – 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 $pdvr \geq 1$ .

15: $ilo$ – Integer * Output

16: $ihi$ – Integer * Output

On exit: ilo and ihi are integer values determined when

A

was balanced. The balanced

A

has

a_{i j} = 0

i > j

and

j = 1, 2, \dots, ilo - 1

i = ihi + 1, \dots, n

17: $scale [\dim]$ – double Output

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

\max (1, n)

On exit: details of the permutations and scaling factors applied when balancing

A

p_{j}

is the index of the row and column interchanged with row and column

j

, and

d_{j}

is the scaling factor applied to row and column

j

, then

$scale [j - 1] = p_{j}$ , for $j = 1, 2, \dots, ilo - 1$ ;
$scale [j - 1] = d_{j}$ , for $j = ilo, \dots, ihi$ ;
$scale [j - 1] = p_{j}$ , for $j = ihi + 1, \dots, n$ .

The order in which the interchanges are made is n to

ihi + 1

, then

1

ilo - 1

18: $abnrm$ – double * Output

On exit: the

1

-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).

19: $rconde [\dim]$ – double Output

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

\max (1, n)

On exit:

rconde [j - 1]

is the reciprocal condition number of the

j

th eigenvalue.

20: $rcondv [\dim]$ – double Output

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

\max (1, n)

On exit:

rcondv [j - 1]

is the reciprocal condition number of the

j

th right eigenvector.

21: $fail$ – 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 or condition numbers have been computed; elements $1$ to $ilo - 1$ and $⟨ value ⟩$ 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

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

f08nbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08nbc 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 f08npc.

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}),

together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.

f08nb: FL CL CPP AD PY MB

NAG CL Interfacef08nbc (dgeevx)

▸▿ 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
f08nbc (dgeevx)