namespace Eigen {

/** \page StructuredMatricesExamples StructuredMatrices examples

These programs solve problems from numerical PDEs, integral equations, signal and image processing, approximation
theory, statistics, control and quantum mechanics with the \ref StructuredMatrices_Module. Each one replaces a dense
or sparse factorization by an algorithm that exploits a structure of the problem -- a Kronecker product or sum,
Toeplitz, Hankel or circulant, Vandermonde, Cauchy, or diagonal plus low rank -- and checks its result against a known
solution or a general-purpose solver. The bracketed numbers cite the module's
\ref StructuredMatricesReferences "references".

\tableofcontents

\section StructuredExamplesGrids Separable operators on tensor-product grids

Finite differences on a tensor-product grid give Kronecker sums of 1-D matrices: with grid values stored with x as the
fast index, the 2-D Laplacian is \f$ D_y \oplus D_x = D_y \otimes I + I \otimes D_x \f$. \ref KroneckerSum applies
such operators without forming them, and \ref BartelsStewart solves them through the Schur forms or
eigendecompositions of the 1-D factors. LeVeque [1] covers the discretizations.

\subsection StructuredExamplePoisson2D The 2-D Poisson equation

The fast Poisson solver: two \f$ n \times n \f$ eigendecompositions instead of a factorization of the
\f$ n^2 \times n^2 \f$ matrix, with the second-order convergence of the discretization: the fast diagonalization
method of Lynch, Rice and Thomas [2].

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredPoisson2D.cpp
</td>
<td>
\verbinclude StructuredPoisson2D.out
</td></tr></table>

\subsection StructuredExampleHeat3D Implicit time stepping of the 3-D heat equation

The decompositions are computed once and reused at every step. A shift such as the identity of
\f$ I + \tau L \f$ goes into one factor, since \f$ I \otimes I \otimes I = I \f$.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredHeat3D.cpp
</td>
<td>
\verbinclude StructuredHeat3D.out
</td></tr></table>

\subsection StructuredExampleConvectionDiffusion Convection-diffusion

Nonsymmetric 1-D operators take the Schur-form path of \ref BartelsStewart, the algorithm of Bartels and Stewart [3];
the solution agrees with \c SparseLU on the assembled matrix.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredConvectionDiffusion.cpp
</td>
<td>
\verbinclude StructuredConvectionDiffusion.out
</td></tr></table>

\subsection StructuredExampleADI Alternating-direction implicit time stepping

The Peaceman-Rachford ADI method [4] treats one direction at a time. Each half step is a \ref KroneckerOperator with an
\c Identity() factor, which costs nothing: the product is a sparse product with the 1-D matrix and the solve one
\c SparseLU of a tridiagonal matrix.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredADI.cpp
</td>
<td>
\verbinclude StructuredADI.out
</td></tr></table>

\subsection StructuredExamplePreconditioner A non-separable potential

When a term breaks the separable structure, the separable part still helps twice: \ref KroneckerSparseView
assembles the sparse matrix in one expression, and the fast Poisson solver preconditions conjugate gradients.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredPreconditioner.cpp
</td>
<td>
\verbinclude StructuredPreconditioner.out
</td></tr></table>

\subsection StructuredExampleTorusHelmholtz Periodic boundaries: the screened Poisson equation on a torus

With periodic boundaries the five-point stencil is a block circulant with circulant blocks, solved by \ref Bccb
through the 2-D FFT [5]. It is also the Kronecker sum of two 1-D periodic second-difference circulants, and the two
solvers agree.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredTorusHelmholtz.cpp
</td>
<td>
\verbinclude StructuredTorusHelmholtz.out
</td></tr></table>

\subsection StructuredExampleSpectralDifferentiation Fourier spectral differentiation

On a periodic grid the spectral differentiation matrices are circulants [21], and their spectra are the wavenumbers:
\ref Circulant::eigenvalues returns \f$ ik \f$, \ref Circulant::rank counts the \f$ N - 2 \f$ nonzero ones (the
constant and the sawtooth mode are annihilated), and the pseudo-inverse \ref Circulant::solve integrates a periodic
function with zero mean. Derivatives and the periodic boundary value problem \f$ u'' - u = f \f$ converge
geometrically for smooth \f$ u \f$.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredSpectralDifferentiation.cpp
</td>
<td>
\verbinclude StructuredSpectralDifferentiation.out
</td></tr></table>

\section StructuredExamplesSpectra Matrix equations and spectra

\subsection StructuredExampleLyapunov The Lyapunov equation of a mass-spring chain

\f$ A X + X A^T = C \f$ is \f$ (A \oplus A)\,\mathrm{vec}(X) = \mathrm{vec}(C) \f$, and the Sylvester equation
\f$ A X + X B = C \f$ is \f$ (B^T \oplus A)\,\mathrm{vec}(X) = \mathrm{vec}(C) \f$. This example computes the
controllability Gramian of a damped chain, whose system matrix has complex eigenvalue pairs; Simoncini [6] surveys the
methods for such matrix equations.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredLyapunov.cpp
</td>
<td>
\verbinclude StructuredLyapunov.out
</td></tr></table>

\subsection StructuredExampleHarmonicOscillator The 2-D quantum harmonic oscillator

A separable Hamiltonian \f$ H_x \oplus H_y \f$ has the pairwise sums of the 1-D energies as its spectrum, which
\ref KroneckerSum::eigenvalues returns without forming the 2-D matrix [7].

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredHarmonicOscillator.cpp
</td>
<td>
\verbinclude StructuredHarmonicOscillator.out
</td></tr></table>

\subsection StructuredExampleImpurity An impurity bound state

A localized perturbation of a diagonal band Hamiltonian is a rank-one update, the Koster-Slater model [8]. \ref DPR1EigenSolver finds the whole
spectrum in \f$ O(N^2) \f$ from the secular equation [9], including the bound state that splits off below the band.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredImpurity.cpp
</td>
<td>
\verbinclude StructuredImpurity.out
</td></tr></table>

\subsection StructuredExampleDivideAndConquer The merge step of divide and conquer

Cuppen's method [25] tears a symmetric tridiagonal matrix into two halves plus a rank-one coupling; after the halves
are diagonalized, what remains is the diagonal-plus-rank-one eigenproblem that \ref DPR1EigenSolver solves from the
secular equation, with eigenvectors orthogonal to working accuracy by the construction of Gu and Eisenstat [9]. The
example performs one such merge and compares with \c SelfAdjointEigenSolver.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredDivideAndConquer.cpp
</td>
<td>
\verbinclude StructuredDivideAndConquer.out
</td></tr></table>

\section StructuredExamplesToeplitz Toeplitz, circulant and Hankel systems

\subsection StructuredExampleLoveEquation Love's integral equation and a preconditioned Toeplitz system

A translation-invariant kernel discretized on a uniform grid gives a \ref Toeplitz matrix. Love's equation [10] is solved
directly by \ref LookAheadLevinson [11] in \f$ O(n^2) \f$. For large systems, conjugate gradients use the FFT-based
Toeplitz product and Strang's \ref Circulant preconditioner [12], [13], \f$ O(n \log n) \f$ per iteration.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredLoveEquation.cpp
</td>
<td>
\verbinclude StructuredLoveEquation.out
</td></tr></table>

\subsection StructuredExamplePade Pade approximation

The denominator of a Pade approximant [14] solves a \ref Hankel system in the Taylor coefficients, solved in
\f$ O(n^2) \f$ through its Toeplitz equivalent. Rescaling the variable equilibrates the coefficients and keeps the
system Hankel.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredPade.cpp
</td>
<td>
\verbinclude StructuredPade.out
</td></tr></table>

\section StructuredExamplesSignals Signal and image processing

\subsection StructuredExampleDeconvolution Deconvolution by Toeplitz least squares

The full convolution of a signal with a kernel is a rectangular \ref Toeplitz matrix. \c LSMR [22] solves the
least-squares problem without forming it: products with the matrix and, through \ref Toeplitz::adjoint, its transpose
cost \f$ O(p \log p) \f$ each by circulant embedding [13]. The matrix-free solution agrees with the dense QR of the
materialized matrix.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredDeconvolution.cpp
</td>
<td>
\verbinclude StructuredDeconvolution.out
</td></tr></table>

\subsection StructuredExampleDeblurring Image deblurring with periodic boundaries

A blur with periodic boundary conditions is a \ref Bccb matrix [5]. A motion blur annihilates whole frequency lines,
which \ref Bccb::rank counts exactly, so the pseudo-inverse \ref Bccb::solve cannot undo it and amplifies the noise
next to those lines. Tikhonov regularization stays within the class: \f$ A^H A + \lambda I \f$ is the Bccb generated by
the autocorrelation of the point spread function, computed with two products, and each \f$ \lambda \f$ costs one
\f$ O(N \log N) \f$ solve.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredDeblurring.cpp
</td>
<td>
\verbinclude StructuredDeblurring.out
</td></tr></table>

\subsection StructuredExampleProny Prony's method

A sum of exponentials obeys a linear recurrence whose coefficients solve a \ref Hankel system in the samples [23]; the
roots of its characteristic polynomial are the damping rates and frequencies, and the amplitudes solve the dual
\ref Vandermonde system with complex nodes, which \ref BjorckPereyra handles in Leja order. The recurrence residual
over all samples is a rectangular Hankel product.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredProny.cpp
</td>
<td>
\verbinclude StructuredProny.out
</td></tr></table>

\section StructuredExamplesApproximation Interpolation and approximation

\subsection StructuredExampleQuadrature Interpolatory quadrature

The weights of an interpolatory quadrature rule solve the transposed \ref Vandermonde system, the dual problem that
\ref BjorckPereyra [16] solves in \f$ O(n^2) \f$; on Chebyshev extreme points they are the Clenshaw-Curtis weights
[15]. The primal solve gives interpolation coefficients.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredQuadrature.cpp
</td>
<td>
\verbinclude StructuredQuadrature.out
</td></tr></table>

\subsection StructuredExampleRationalApproximation A discretized Cauchy integral

Representing an analytic function by point charges outside its domain leads to a \ref Cauchy system, factored with
pivoting in \f$ O(n^2) \f$ by \ref CauchyLU [18]; the error decays geometrically, as for the trapezoidal rule on a
circle [17].

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredRationalApproximation.cpp
</td>
<td>
\verbinclude StructuredRationalApproximation.out
</td></tr></table>

\subsection StructuredExamplePolynomialFit Polynomial least squares and the Hilbert matrix

The normal equations of continuous least-squares approximation in the monomial basis have the Hilbert matrix, a
\ref Cauchy matrix [26]: \ref CauchyLU solves them in \f$ O(n^2) \f$ and \ref Cauchy::determinant gives
\f$ \det H_n \f$ in closed form where dense LU loses digits. The fitted polynomial is far less sensitive to
\f$ \kappa(H_n) \f$ than its coefficients, and matches the discrete fit by a dense QR of the rectangular
\ref Vandermonde matrix; Vandermonde products evaluate both.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredPolynomialFit.cpp
</td>
<td>
\verbinclude StructuredPolynomialFit.out
</td></tr></table>

\subsection StructuredExampleSeparableLeastSquares Separable least squares on a grid

The design matrix of a tensor-product polynomial fit on a grid is the Kronecker product of two rectangular Vandermonde
matrices. \ref KroneckerOperator::leastSquaresSolve applies \f$ (A \otimes B)^+ = A^+ \otimes B^+ \f$ [27] with one
complete orthogonal decomposition per factor, and \ref KroneckerOperator::rank and
\ref KroneckerOperator::singularValues come from the factors too; the dense decomposition of the assembled matrix
agrees.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredSeparableLeastSquares.cpp
</td>
<td>
\verbinclude StructuredSeparableLeastSquares.out
</td></tr></table>

\section StructuredExamplesStatistics Statistics and estimation

\subsection StructuredExampleGaussianProcess A Gaussian-process likelihood

A covariance that is noise plus a low-rank feature model [19], [20] is a \ref DiagonalPlusLowRank operator. Its solve
(Woodbury identity) and determinant (matrix determinant lemma) cost \f$ O(n k^2) \f$, the price of evaluating the
log-likelihood.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredGaussianProcess.cpp
</td>
<td>
\verbinclude StructuredGaussianProcess.out
</td></tr></table>

\subsection StructuredExampleKalmanUpdate A Kalman measurement update

With a diagonal prior and \f$ k \f$ measurements, the posterior precision is diagonal plus rank \f$ k \f$, and
\ref DiagonalPlusLowRank::inverse returns the posterior covariance as another \ref DiagonalPlusLowRank, since the
Woodbury identity is the Kalman update [24]. The posterior mean is one \f$ O(nk) \f$ product and the posterior variances are
read off the diagonal; the dense \f$ n \times n \f$ computation agrees.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include StructuredKalmanUpdate.cpp
</td>
<td>
\verbinclude StructuredKalmanUpdate.out
</td></tr></table>

*/

}  // namespace Eigen