NAG Library Function Document

nag_dgeev (f08nac)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{order}$ – Nag_OrderTypeInput

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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:    $\mathbf{jobvl}$ – Nag_LeftVecsTypeInput

On entry: if $\mathbf{jobvl}=\mathrm{Nag\_NotLeftVecs}$, the left eigenvectors of $A$ are not computed.

If $\mathbf{jobvl}=\mathrm{Nag\_LeftVecs}$, the left eigenvectors of $A$ are computed.

Constraint: $\mathbf{jobvl}=\mathrm{Nag\_NotLeftVecs}$ or $\mathrm{Nag\_LeftVecs}$.

3:    $\mathbf{jobvr}$ – Nag_RightVecsTypeInput

On entry: if $\mathbf{jobvr}=\mathrm{Nag\_NotRightVecs}$, the right eigenvectors of $A$ are not computed.

If $\mathbf{jobvr}=\mathrm{Nag\_RightVecs}$, the right eigenvectors of $A$ are computed.

Constraint: $\mathbf{jobvr}=\mathrm{Nag\_NotRightVecs}$ or $\mathrm{Nag\_RightVecs}$.

4:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

5:    $\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array a must be at least $\max \left(1,\mathbf{pda}\times \mathbf{n}\right)$.

The $\left(i,j\right)$th element of the matrix $A$ is stored in

• $\mathbf{a}\left[\left(j-1\right)\times \mathbf{pda}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{a}\left[\left(i-1\right)\times \mathbf{pda}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the $n$ by $n$ matrix $A$.

On exit: a has been overwritten.

6:    $\mathbf{pda}$ – IntegerInput

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

Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

7:    $\mathbf{wr}\left[\mathit{dim}\right]$ – doubleOutput

8:    $\mathbf{wi}\left[\mathit{dim}\right]$ – doubleOutput

Note: the dimension, dim, of the arrays wr and wi must be at least $\max \left(1,\mathbf{n}\right)$.

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:    $\mathbf{vl}\left[\mathit{dim}\right]$ – doubleOutput

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

• $\max \left(1,\mathbf{pdvl}\times \mathbf{n}\right)$ when $\mathbf{jobvl}=\mathrm{Nag\_LeftVecs}$;
• $1$ otherwise.

Where $\mathbf{VL}\left(i,j\right)$ appears in this document, it refers to the array element

• $\mathbf{vl}\left[\left(j-1\right)\times \mathbf{pdvl}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{vl}\left[\left(i-1\right)\times \mathbf{pdvl}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: if $\mathbf{jobvl}=\mathrm{Nag\_LeftVecs}$, 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 $u_j=\mathbf{VL}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then $u_j=\mathbf{VL}\left(\mathit{i},j\right)+\mathit{i}\times \mathbf{VL}\left(\mathit{i},j+1\right)$ and $u_{j+1}=\mathbf{VL}\left(\mathit{i},j\right)-\mathit{i}\times \mathbf{VL}\left(\mathit{i},j+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

If $\mathbf{jobvl}=\mathrm{Nag\_NotLeftVecs}$, vl is not referenced.

10:  $\mathbf{pdvl}$ – IntegerInput

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

Constraints:

• if $\mathbf{jobvl}=\mathrm{Nag\_LeftVecs}$, $\mathbf{pdvl}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{pdvl}\ge 1$.

11:  $\mathbf{vr}\left[\mathit{dim}\right]$ – doubleOutput

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

• $\max \left(1,\mathbf{pdvr}\times \mathbf{n}\right)$ when $\mathbf{jobvr}=\mathrm{Nag\_RightVecs}$;
• $1$ otherwise.

Where $\mathbf{VR}\left(i,j\right)$ appears in this document, it refers to the array element

• $\mathbf{vr}\left[\left(j-1\right)\times \mathbf{pdvr}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{vr}\left[\left(i-1\right)\times \mathbf{pdvr}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: if $\mathbf{jobvr}=\mathrm{Nag\_RightVecs}$, 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 $v_j=\mathbf{VR}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then $v_j=\mathbf{VR}\left(\mathit{i},j\right)+\mathit{i}\times \mathbf{VR}\left(\mathit{i},j+1\right)$ and $v_{j+1}=\mathbf{VR}\left(\mathit{i},j\right)-\mathit{i}\times \mathbf{VR}\left(\mathit{i},j+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

If $\mathbf{jobvr}=\mathrm{Nag\_NotRightVecs}$, vr is not referenced.

12:  $\mathbf{pdvr}$ – IntegerInput

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

Constraints:

• if $\mathbf{jobvr}=\mathrm{Nag\_RightVecs}$, $\mathbf{pdvr}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{pdvr}\ge 1$.

13:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_CONVERGENCE

The $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $〈\mathit{\text{value}}〉$ to n of wr and wi contain eigenvalues which have converged.

NE_ENUM_INT_2

On entry, $\mathbf{jobvl}=〈\mathit{\text{value}}〉$, $\mathbf{pdvl}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{jobvl}=\mathrm{Nag\_LeftVecs}$, $\mathbf{pdvl}\ge \max \left(1,\mathbf{n}\right)$;
otherwise $\mathbf{pdvl}\ge 1$.

On entry, $\mathbf{jobvr}=〈\mathit{\text{value}}〉$, $\mathbf{pdvr}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{jobvr}=\mathrm{Nag\_RightVecs}$, $\mathbf{pdvr}\ge \max \left(1,\mathbf{n}\right)$;
otherwise $\mathbf{pdvr}\ge 1$.

NE_INT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}>0$.

On entry, $\mathbf{pdvl}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdvl}>0$.

On entry, $\mathbf{pdvr}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdvr}>0$.

NE_INT_2

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08nace.c)

10.2

Program Data

Program Data (f08nace.d)

10.3

Program Results

Program Results (f08nace.r)