NAG FL Interface
f12jtf (feast_​complex_​gen_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{handle}$ – Type (c_ptr) Input

On entry: the handle to the internal data structure used by the NAG FEAST suite. It needs to be initialized by f12jaf. It must not be changed between calls to the NAG FEAST suite.

2: $\mathbf{irevcm}$ – Integer Input/Output

On initial entry: $\mathbf{irevcm}=0$, otherwise an error condition will be raised.

On intermediate re-entry: must be unchanged from its previous exit value. Changing irevcm to any other value between calls will result in an error.

On intermediate exit: has the following meanings.

$\mathbf{irevcm}=1$

The calling program must compute a factorization of the matrix $\mathbf{ze}B-A$ suitable for solving a linear system, for example using f11dnf which computes an incomplete $LU$ factorization of a complex sparse matrix. All arguments to the routine must remain unchanged.

Note:  the factorization can be computed in single precision.

$\mathbf{irevcm}=2$

The calling program must compute the solution to the linear system $(\mathbf{ze}B-A)w=y$, overwriting $y$ with the result $w$. The matrix $\mathbf{ze}B-A$ has previously been factorized (when $\mathbf{irevcm}=1$ was returned) and this factorization can be reused here.

Note:  the solve can be performed in single precision.

$\mathbf{irevcm}=3$

Optionally, the calling program must compute a factorization of the matrix ${(\mathbf{ze}B-A)}^{\mathrm{T}}$. This need only be done if it is not possible to use the factorization computed when $\mathbf{irevcm}=1$ was returned to solve linear systems involving ${(\mathbf{ze}B-A)}^{\mathrm{T}}$. If this factorization is to be computed, then the factorization from $\mathbf{irevcm}=1$ must not be overwritten.

Note:  the factorization can be performed in single precision.

$\mathbf{irevcm}=4$

The calling program must compute the solution to the linear system ${(\mathbf{ze}B-A)}^{\mathrm{T}}w=y$, overwriting $y$ with the result $w$. If it is not possible to use the factorization of $\mathbf{ze}B-A$ (computed when $\mathbf{irevcm}=1$ was returned) then the factorization of ${(\mathbf{ze}B-A)}^{\mathrm{T}}$ (computed when $\mathbf{irevcm}=2$ was returned) should be used here.

Note:  the solve can be performed in single precision.

$\mathbf{irevcm}=5$

The calling program must compute $Az$, storing the result in $x$.

$\mathbf{irevcm}=6$

The calling program must compute $A^{\mathrm{T}}z$, storing the result in $x$.

$\mathbf{irevcm}=7$

The calling program must compute $Bz$, storing the result in $x$. If a standard eigenproblem is being solved (so that $B=I$) then the calling program should set $x=z$.

$\mathbf{irevcm}=8$

The calling program must compute $B^{\mathrm{T}}z$, storing the result in $x$. If a standard eigenproblem is being solved (so that $B=I$) then the calling program should set $x=z$.

On final exit: $\mathbf{irevcm}=0$: f12jtf has completed its tasks. The value of ifail determines whether the iteration has been successfully completed, or whether errors have been detected.

Constraint: on initial entry, $\mathbf{irevcm}=0$; on re-entry irevcm must remain unchanged.

Note: the matrices $x$, $y$ and $z$ referred to in this section are all of size $\mathbf{n}\times \mathbf{m0}$ and are stored in the arrays x, y and z, respectively.

Note: any values you return to f12jtf as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by f12jtf. If your code does inadvertently return any NaNs or infinities, f12jtf is likely to produce unexpected results.

3: $\mathbf{ze}$ – Complex (Kind=nag_wp) Input/Output

On initial entry: need not be set.

On intermediate exit: contains the current point on the contour.

• If $\mathbf{irevcm}=1$, then this must be used by the calling program to form a factorization of the matrix $\mathbf{ze}B-A$.
• If $\mathbf{irevcm}=3$, then, optionally, this can be used to form a factorization of ${(\mathbf{ze}B-A)}^{\mathrm{H}}$.

4: $\mathbf{n}$ – Integer Input

On entry: the order of the matrix $A$ (and the order of the matrix $B$ for the generalized problem) that defines the eigenvalue problem.

Constraint: $\mathbf{n}\ge 1$.

5: $\mathbf{x}(\mathbf{ldx},*)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array x must be at least $2\times \mathbf{m0}$.

On initial entry: need not be set.

On intermediate exit:

• if $\mathbf{irevcm}=5$, the calling program must compute $Az$, storing the result in $x$ prior to re-entry.
• If $\mathbf{irevcm}=6$, the calling program must compute $A^{\mathrm{H}}z$, storing the result in $x$ prior to re-entry.
• If $\mathbf{irevcm}=7$, the calling program must compute $Bz$, storing the result in $x$ prior to re-entry.
• If $\mathbf{irevcm}=8$, the calling program must compute $B^{\mathrm{H}}z$, storing the result in $x$ prior to re-entry.

Note: the matrices $x$ and $z$ are stored in the first m0 columns of the arrays x and z, respectively.

6: $\mathbf{ldx}$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which f12jtf is called.

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

7: $\mathbf{y}(\mathbf{ldy},*)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array y must be at least $\mathbf{m0}$.

On initial entry: need not be set.

On intermediate exit:

• if $\mathbf{irevcm}=2$, the calling program must compute the solution to the linear system $(\mathbf{ze}B-A)w=y$, overwriting $y$ with the result $w$, prior to re-entry. The linear system has m0 right-hand sides.
• If $\mathbf{irevcm}=4$, the calling program must compute the solution to the linear system ${(\mathbf{ze}B-A)}^{\mathrm{H}}w=y$, overwriting $y$ with the result $w$, prior to re-entry. The linear system has m0 right-hand sides.

8: $\mathbf{ldy}$ – Integer Input

On entry: the first dimension of the array y as declared in the (sub)program from which f12jtf is called.

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

9: $\mathbf{m0}$ – Integer Input/Output

On initial entry: the size of the search subspace used to find the eigenvalues. This should exceed the number of eigenvalues within the search contour. See Section 9 for further details.

On intermediate re-entry: m0 must remain unchanged.

On exit: if the initial search subspace was found by f12jtf to be too large, then a new smaller suitable choice is returned.

Constraint: $0<\mathbf{m0}\le \mathbf{n}$.

10: $\mathbf{nconv}$ – Integer Output

On exit: the number of eigenvalues found within the search contour.

Note: if the optional parameter $\mathbf{Execution\; Mode}=\mathrm{Estimate}$ was set in the option setting routine f12jbf, then nconv contains a stochastic estimate of the number of eigenvalues within the search contour.

11: $\mathbf{d}\left(*\right)$ – Complex (Kind=nag_wp) array Input/Output

Note: the dimension of the array d must be at least $\mathbf{m0}$.

On initial entry: if the option $\mathbf{Subspace}=\mathrm{Yes}$ was set using the option setting routine f12jbf, then d should contain an initial guess at the eigenvalues lying within the eigenvector search subspace (this subspace should be specified by z), otherwise d need not be set.

On final exit: the first nconv entries in d contain the eigenvalues.

Note: if the option $\mathbf{Subspace}=\mathrm{Yes}$ was set using the option setting routine f12jbf, then on final exit d contains an estimate of the eigenvalues after a single contour integral.

12: $\mathbf{z}(\mathbf{ldz},*)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array z must be at least $2\times \mathbf{m0}$.

On initial entry: if the option $\mathbf{Subspace}=\mathrm{Yes}$ was set using the option setting routine f12jbf, then z should contain an initial guess at the eigenvector search subspace, otherwise z need not be set.

Note: if the option $\mathbf{Subspace}=\mathrm{Yes}$ was set but only right eigenvectors are to be computed, then only the first m0 columns of z need to be set. However, if the optional parameter $\mathbf{Eigenvectors}=\mathrm{Two-sided}$ was set using f12jbf, so that both left and right eigenvectors are returned, then the first m0 columns of z should contain the initial guess for the right eigenvector search subspace and the remaining m0 columns should contain an initial guess for the left eigenvector search subspace.

On intermediate exit: must not be changed.

On final exit: the first nconv columns of z contain the right eigenvectors corresponding to the eigenvalues found within the contour. If the option $\mathbf{Eigenvectors}=\mathrm{Two-sided}$ was set in the option setting routine f12jbf, the columns $\mathbf{m0}+1:\mathbf{m0}+\mathbf{nconv}$ of z contain the left eigenvectors corresponding to the eigenvalues found within the contour.

Note: if the option $\mathbf{Execution\; Mode}=\mathrm{Subspace}$ was set using the option setting routine f12jbf, then on final exit columns $1:\mathbf{m0}$ of z contain the current search subspace after one contour integral.

13: $\mathbf{ldz}$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f12jtf is called.

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

14: $\mathbf{eps}$ – Real (Kind=nag_wp) Input/Output

On initial entry: need not be set.

On exit: the relative error on the trace. At iteration $k$, eps is given by the expression $|{\mathrm{trace}}_k-{\mathrm{trace}}_{k-1}|/(\left|E_{\mathrm{mid}}\right|+r)$, where ${\mathrm{trace}}_k$ is the sum of the eigenvalues found at the $k$th iteration, $E_{\mathrm{mid}}$ is the centre of mass of the contour and $r$ is the radius of the circle, centred at $E_{\mathrm{mid}}$ which just encloses the contour.

15: $\mathbf{iter}$ – Integer Input/Output

On initial entry: need not be set.

On exit: the number of subspace iterations performed.

16: $\mathbf{resid}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension of the array resid must be at least $2\times \mathbf{m0}$.

On initial entry: need not be set.

On final exit: for $i=1,\dots ,\mathbf{nconv}$, $\mathbf{resid}\left(i\right)$ contains the relative residual, in the $1$-norm, of the $i$th eigenpair found, that is $\mathbf{resid}\left(i\right)=\Vert A{z}_i-{\lambda }_{i}B{z}_i\Vert /(\Vert B{z}_i\Vert \times (\left|E_{\mathrm{mid}}\right|+r))$, where $E_{\mathrm{mid}}$ is the centre of mass of the contour and $r$ is the radius of the circle, centred at $E_{\mathrm{mid}}$ which just encloses the contour.

If the optional parameter $\mathbf{Eigenvectors}=\mathrm{Two-sided}$ was set in the option setting routine f12jbf, the entries $\mathbf{m0}+1:\mathbf{m0}+\mathbf{nconv}$ of resid contain corresponding residuals for the left eigenvectors, $\mathbf{resid}\left(i\right)=\Vert A^{\mathrm{H}}{z}_i-{\overline{\lambda }}_i{B}^{\mathrm{H}}{z}_i\Vert /(\Vert B^{\mathrm{H}}{z}_i\Vert \times (\left|E_{\mathrm{mid}}\right|+r))$.

17: $\mathbf{ifail}$ – Integer Input/Output

On initial entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when $\mathbf{ifail}\ne \mathbf{0}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On final exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

Either one of the contour setting routines f12jff or f12jgf has not been called prior to the first call of this routine or the supplied handle has become corrupted.

$\mathbf{ifail}=2$

No eigenvalues were found within the search contour.

$\mathbf{ifail}=3$

The routine did not converge after the maximum number of iterations. The results returned may still be useful, however they might be improved by increasing the maximum number of iterations using the option setting routine f12jbf, increasing the size of the eigenvector search subspace, m0, or experimenting with the choice of contour. Note that the returned eigenvalues and eigenvectors, together with the returned value of m0, can be used as the initial estimates for a new iteration of the solver.

$\mathbf{ifail}=4$

The size of the eigenvector search subspace, m0, is too small.

$\mathbf{ifail}=5$

The optional parameter $\mathbf{Execution\; Mode}=\mathrm{Subspace}$ was set using f12jbf. Columns $1:\mathbf{m0}$ of z contain the search subspace after one contour integral and d contains an estimate of the eigenvalues.

$\mathbf{ifail}=6$

The optional parameter $\mathbf{Execution\; Mode}=\mathrm{Estimate}$ was set using f12jbf so only a stochastic estimate of the number of eigenvalues within the contour has been returned.

$\mathbf{ifail}=7$

An internal error occurred in the reduced eigenvalue solver. Please contact NAG.

$\mathbf{ifail}=8$

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 1$.

$\mathbf{ifail}=9$

On entry, $\mathbf{m0}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $0<\mathbf{m0}\le \mathbf{n}$.

$\mathbf{ifail}=10$

On entry, $\mathbf{ldx}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ldx}\ge \mathbf{n}$.

$\mathbf{ifail}=11$

On entry, $\mathbf{ldy}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ldy}\ge \mathbf{n}$.

$\mathbf{ifail}=12$

On entry, $\mathbf{ldz}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ldz}\ge \mathbf{n}$.

$\mathbf{ifail}=13$

On initial entry, $\mathbf{irevcm}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{irevcm}=0$.

On intermediate entry, $\mathbf{irevcm}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{irevcm}=1$, $2$, $3$, $4$, $5$, $6$, $7$ or $8$.

$\mathbf{ifail}=14$

The routine converged but the left/right eigenvector subspaces are not bi-orthonormal.

The routine converged but the right eigenvector subspace is not orthonormal.

$\mathbf{ifail}=15$

The option $\mathbf{Subspace}=\mathrm{Yes}$ was set using the option setting routine f12jbf but no nonzero elements were found in the supplied subspace.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Additional Licensor

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results