## 1Purpose

f02jqc solves the quadratic eigenvalue problem
 $λ2 ⁢ A + λ ⁢ B + C ⁢ x = 0 ,$
where $A$, $B$ and $C$ are complex $n$ by $n$ matrices.
The function returns the $2n$ eigenvalues, ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,2n$, and can optionally return the corresponding right eigenvectors, ${x}_{j}$ and/or left eigenvectors, ${y}_{j}$ as well as estimates of the condition numbers of the computed eigenvalues and backward errors of the computed right and left eigenvectors. A left eigenvector satisfies the equation
 $yH ⁢ λ2⁢A+ λ⁢B+ C = 0 ,$
where ${y}^{\mathrm{H}}$ is the complex conjugate transpose of $y$.
$\lambda$ is represented as the pair $\left(\alpha ,\beta \right)$, such that $\lambda =\alpha /\beta$. Note that the computation of $\alpha /\beta$ may overflow and indeed $\beta$ may be zero.

## 2Specification

 #include
 void f02jqc (Nag_ScaleType scal, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_CondErrType sense, double tol, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, Complex c[], Integer pdc, Complex alpha[], Complex beta[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, double s[], double bevl[], double bevr[], Integer *iwarn, NagError *fail)
The function may be called by the names: f02jqc or nag_eigen_complex_gen_quad.

## 3Description

The quadratic eigenvalue problem is solved by linearizing the problem and solving the resulting $2n$ by $2n$ generalized eigenvalue problem. The linearization is chosen to have favourable conditioning and backward stability properties. An initial preprocessing step is performed that reveals and deflates the zero and infinite eigenvalues contributed by singular leading and trailing matrices.
The algorithm is backward stable for problems that are not too heavily damped, that is $‖B‖\le 10\sqrt{‖A‖·‖C‖}$.
Further details on the algorithm are given in Hammarling et al. (2013).

## 4References

Fan H -Y, Lin W.-W and Van Dooren P. (2004) Normwise scaling of second order polynomial matrices SIAM J. Matrix Anal. Appl. 26, 1 252–256
Gaubert S and Sharify M (2009) Tropical scaling of polynomial matrices Lecture Notes in Control and Information Sciences Series 389 291–303 Springer–Verlag
Hammarling S, Munro C J and Tisseur F (2013) An algorithm for the complete solution of quadratic eigenvalue problems ACM Trans. Math. Software. 39(3):18:1–18:119 http://eprints.maths.manchester.ac.uk/2061/

## 5Arguments

1: $\mathbf{scal}$Nag_ScaleType Input
On entry: determines the form of scaling to be performed on $A$, $B$ and $C$.
${\mathbf{scal}}=\mathrm{Nag_NoScale}$
No scaling.
${\mathbf{scal}}=\mathrm{Nag_CondFanLinVanDooren}$ (the recommended value)
Fan, Lin and Van Dooren scaling if $\frac{‖B‖}{\sqrt{‖A‖×‖C‖}}<10$ and no scaling otherwise where $‖Z‖$ is the Frobenius norm of $Z$.
${\mathbf{scal}}=\mathrm{Nag_FanLinVanDooren}$
Fan, Lin and Van Dooren scaling.
${\mathbf{scal}}=\mathrm{Nag_TropicalLargest}$
Tropical scaling with largest root.
${\mathbf{scal}}=\mathrm{Nag_TropicalSmallest}$
Tropical scaling with smallest root.
Constraint: ${\mathbf{scal}}=\mathrm{Nag_NoScale}$, $\mathrm{Nag_CondFanLinVanDooren}$, $\mathrm{Nag_FanLinVanDooren}$, $\mathrm{Nag_TropicalLargest}$ or $\mathrm{Nag_TropicalSmallest}$.
2: $\mathbf{jobvl}$Nag_LeftVecsType Input
On entry: if ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, do not compute left eigenvectors.
If ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, compute the left eigenvectors.
If ${\mathbf{sense}}=\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrLeft}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$, jobvl must be set to $\mathrm{Nag_LeftVecs}$.
Constraint: ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$ or $\mathrm{Nag_LeftVecs}$.
3: $\mathbf{jobvr}$Nag_RightVecsType Input
On entry: if ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, do not compute right eigenvectors.
If ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, compute the right eigenvectors.
If ${\mathbf{sense}}=\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrRight}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$, jobvr must be set to $\mathrm{Nag_RightVecs}$.
Constraint: ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$ or $\mathrm{Nag_RightVecs}$.
4: $\mathbf{sense}$Nag_CondErrType Input
On entry: determines whether, or not, condition numbers and backward errors are computed.
${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$
Do not compute condition numbers, or backward errors.
${\mathbf{sense}}=\mathrm{Nag_CondOnly}$
Just compute condition numbers for the eigenvalues.
${\mathbf{sense}}=\mathrm{Nag_BackErrLeft}$
Just compute backward errors for the left eigenpairs.
${\mathbf{sense}}=\mathrm{Nag_BackErrRight}$
Just compute backward errors for the right eigenpairs.
${\mathbf{sense}}=\mathrm{Nag_BackErrBoth}$
Compute backward errors for the left and right eigenpairs.
${\mathbf{sense}}=\mathrm{Nag_CondBackErrLeft}$
Compute condition numbers for the eigenvalues and backward errors for the left eigenpairs.
${\mathbf{sense}}=\mathrm{Nag_CondBackErrRight}$
Compute condition numbers for the eigenvalues and backward errors for the right eigenpairs.
${\mathbf{sense}}=\mathrm{Nag_CondBackErrBoth}$
Compute condition numbers for the eigenvalues and backward errors for the left and right eigenpairs.
Constraint: ${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$, $\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrLeft}$, $\mathrm{Nag_BackErrRight}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$.
5: $\mathbf{tol}$double Input
On entry: tol is used as the tolerance for making decisions on rank in the deflation procedure. If tol is zero on entry then is used in place of tol, where machine precision is as returned by function X02AJC. A diagonal element of a triangular matrix, $R$, is regarded as zero if $\left|{r}_{jj}\right|\le {\mathbf{tol}}×\mathrm{size}\left(X\right)$, or when tol is zero, where $\mathrm{size}\left(X\right)$ is based on the size of the absolute values of the elements of the matrix $X$ containing the matrix $R$. See Hammarling et al. (2013) for the motivation. If tol is $-1.0$ on entry then no deflation is attempted. The recommended value for tol is zero.
6: $\mathbf{n}$Integer Input
On entry: the order of the matrices $A$, $B$ and $C$.
Constraint: ${\mathbf{n}}\ge 0$.
7: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: a is used as internal workspace, but if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$ or ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, a is restored on exit.
8: $\mathbf{pda}$Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
9: $\mathbf{b}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array b must be at least ${\mathbf{pdb}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $B$.
On exit: b is used as internal workspace, but is restored on exit.
10: $\mathbf{pdb}$Integer Input
On entry: the stride separating matrix row elements in the array b.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
11: $\mathbf{c}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array c must be at least ${\mathbf{pdc}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $C$.
On exit: c is used as internal workspace, but if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$ or ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, c is restored on exit.
12: $\mathbf{pdc}$Integer Input
On entry: the stride separating matrix row elements in the array c.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{n}}$.
13: $\mathbf{alpha}\left[2×{\mathbf{n}}\right]$Complex Output
On exit: ${\mathbf{alpha}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,2n$, contains the first part of the the $j$th eigenvalue pair $\left({\alpha }_{j},{\beta }_{j}\right)$ of the quadratic eigenvalue problem.
14: $\mathbf{beta}\left[2×{\mathbf{n}}\right]$Complex Output
On exit: ${\mathbf{beta}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,2n$, contains the second part of the $j$th eigenvalue pair $\left({\alpha }_{j},{\beta }_{j}\right)$ of the quadratic eigenvalue problem. Although beta is declared complex, it is actually real and non-negative. Infinite eigenvalues have ${\beta }_{j}$ set to zero.
15: $\mathbf{vl}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array vl must be at least $2×{\mathbf{n}}$ when ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$.
where ${\mathbf{VL}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{vl}}\left[\left(j-1\right)×{\mathbf{pdvl}}+i-1\right]$.
On exit: if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, the left eigenvectors ${y}_{j}$ are stored one after another in vl, in the same order as the corresponding eigenvalues. Each eigenvector will be normalized with length unity and with the element of largest modulus real and positive.
If ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, vl is not referenced and may be NULL.
16: $\mathbf{pdvl}$Integer Input
On entry: the stride separating matrix row elements in the array vl.
Constraint: ${\mathbf{pdvl}}\ge {\mathbf{n}}$.
17: $\mathbf{vr}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array vr must be at least $2×{\mathbf{n}}$ when ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$.
where ${\mathbf{VR}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{vr}}\left[\left(j-1\right)×{\mathbf{pdvr}}+i-1\right]$.
On exit: if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, the right eigenvectors ${x}_{j}$ are stored one after another in vr, in the same order as the corresponding eigenvalues. Each eigenvector will be normalized with length unity and with the element of largest modulus real and positive.
If ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, vr is not referenced and may be NULL.
18: $\mathbf{pdvr}$Integer Input
On entry: the stride separating matrix row elements in the array vr.
Constraint: ${\mathbf{pdvr}}\ge {\mathbf{n}}$.
19: $\mathbf{s}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array s must be at least
• $2×{\mathbf{n}}$ when ${\mathbf{sense}}=\mathrm{Nag_CondOnly}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$.
also: computing the condition numbers of the eigenvalues requires that both the left and right eigenvectors be computed.
On exit: if ${\mathbf{sense}}=\mathrm{Nag_CondOnly}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$, ${\mathbf{s}}\left[j-1\right]$ contains the condition number estimate for the $j$th eigenvalue (large condition numbers imply that the problem is near one with multiple eigenvalues). Infinite condition numbers are returned as the largest model real number (X02ALC).
If ${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$, $\mathrm{Nag_BackErrLeft}$, $\mathrm{Nag_BackErrRight}$ or $\mathrm{Nag_BackErrBoth}$, s is not referenced and may be NULL.
20: $\mathbf{bevl}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array bevl must be at least
• $2×{\mathbf{n}}$ when ${\mathbf{sense}}=\mathrm{Nag_BackErrLeft}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$ or $\mathrm{Nag_CondBackErrBoth}$.
On exit: if ${\mathbf{sense}}=\mathrm{Nag_BackErrLeft}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$ or $\mathrm{Nag_CondBackErrBoth}$, ${\mathbf{bevl}}\left[j-1\right]$ contains the backward error estimate for the computed left eigenpair $\left({\lambda }_{j},{y}_{j}\right)$.
If ${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$, $\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrRight}$ or $\mathrm{Nag_CondBackErrRight}$, bevl is not referenced and may be NULL.
21: $\mathbf{bevr}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array bevr must be at least
• $2×{\mathbf{n}}$ when ${\mathbf{sense}}=\mathrm{Nag_BackErrRight}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$.
On exit: if ${\mathbf{sense}}=\mathrm{Nag_BackErrRight}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$, ${\mathbf{bevr}}\left[j-1\right]$ contains the backward error estimate for the computed right eigenpair $\left({\lambda }_{j},{x}_{j}\right)$.
If ${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$, $\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrLeft}$ or $\mathrm{Nag_CondBackErrLeft}$, bevr is not referenced and may be NULL.
22: $\mathbf{iwarn}$Integer * Output
On exit: iwarn will be positive if there are warnings, otherwise iwarn will be $0$.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR then:
• if ${\mathbf{iwarn}}=1$ then one, or both, of the matrices $A$ and $C$ is zero. In this case no scaling is performed, even if ${\mathbf{scal}}=\mathrm{Nag_CondFanLinVanDooren}$;
• if ${\mathbf{iwarn}}=2$ then the matrices $A$ and $C$ are singular, or nearly singular, so the problem is potentially ill-posed;
• if ${\mathbf{iwarn}}=3$ then both the conditions for ${\mathbf{iwarn}}=1$ and ${\mathbf{iwarn}}=2$ above, apply. If ${\mathbf{iwarn}}=4$, $‖{\mathbf{b}}‖\ge 10\sqrt{‖A‖·‖C‖}$ and backward stability cannot be guaranteed.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_ITERATIONS_QZ, f08xnc has flagged that iwarn eigenvalues are invalid.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_ITERATIONS_QZ, f08wnc has flagged that iwarn eigenvalues are invalid.
23: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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_ARRAY_SIZE
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdvl}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{jobvl}}=〈\mathit{\text{value}}〉$.
Constraint: when ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvl}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdvr}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{jobvr}}=〈\mathit{\text{value}}〉$.
Constraint: when ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvr}}\ge {\mathbf{n}}$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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_INVALID_VALUE
On entry, ${\mathbf{sense}}=〈\mathit{\text{value}}〉$ and ${\mathbf{jobvl}}=〈\mathit{\text{value}}〉$.
Constraint: when ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, ${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$ or $\mathrm{Nag_BackErrRight}$,
when ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{sense}}=\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrLeft}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$.
On entry, ${\mathbf{sense}}=〈\mathit{\text{value}}〉$ and ${\mathbf{jobvr}}=〈\mathit{\text{value}}〉$.
Constraint: when ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, ${\mathbf{sense}}=\mathrm{Nag_NoCondBackErr}$ or $\mathrm{Nag_BackErrLeft}$,
when ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{sense}}=\mathrm{Nag_CondOnly}$, $\mathrm{Nag_BackErrRight}$, $\mathrm{Nag_BackErrBoth}$, $\mathrm{Nag_CondBackErrLeft}$, $\mathrm{Nag_CondBackErrRight}$ or $\mathrm{Nag_CondBackErrBoth}$.
NE_ITERATIONS_QZ
The $QZ$ iteration failed in f08wnc.
iwarn returns the value of info returned by f08wnc. This failure is unlikely to happen, but if it does, please contact NAG.
The $QZ$ iteration failed in f08xnc.
iwarn returns the value of info returned by f08xnc. This failure is unlikely to happen, but if it does, please contact NAG.
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_SINGULAR
The quadratic matrix polynomial is nonregular (singular).

## 7Accuracy

The algorithm is backward stable for problems that are not too heavily damped, that is $‖B‖\le \sqrt{‖A‖·‖C‖}$.

## 8Parallelism and Performance

f02jqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f02jqc 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.

None.

## 10Example

To solve the quadratic eigenvalue problem
 $λ2 A + λB + C x = 0$
where
 $A = 2i 4i 4i 6i 2i 2i 6i 4i 2i , B = 3+3i 2+2i 1+1i 2+2i 1+1i 3+3i 1+1i 3+3i 2+2i and C = 1 1 2 2 3 1 3 1 2 .$
The example also returns the left eigenvectors, condition numbers for the computed eigenvalues and the maximum backward errors of the computed right and left eigenpairs.

### 10.1Program Text

Program Text (f02jqce.c)

### 10.2Program Data

Program Data (f02jqce.d)

### 10.3Program Results

Program Results (f02jqce.r)