Rayleigh–Ritz method

The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.

In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression, on which we can use an eigenvalue algorithm.

It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. In quantum mechanics, where a system of particles is described using a Hamiltonian, the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy. In the finite element method context, mathematically the same algorithm is commonly called the Ritz-Galerkin method. The Rayleigh–Ritz method or Ritz method terminology is typical in mechanical and structural engineering to approximate the eigenmodes and resonant frequencies of a structure.

Naming and attribution

The name of the method and its origin story have been debated by histroians. It has been called Ritz method after Walther Ritz, since the numerical procedure has been published by Walther Ritz in 1908-1909. According to A. W. Leissa, Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the Rayleigh quotient make the case for the name Rayleigh–Ritz method. According to S. Ilanko, citing Richard Courant, both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined; see the article Ritz method for details. Ironically for the debate, the modern justification of the algorithm drops the calculus of variations in favor of the simpler and more general approach of orthogonal projection as in Galerkin method named after Boris Galerkin, thus leading also to the Ritz-Galerkin method naming.

Method

Let $T$ be a linear operator on a Hilbert space ${\mathcal {H}}$, with inner product $(\cdot ,\cdot )$. Now consider a finite set of functions ${\mathcal {L}}=\{\varphi _{1},...,\varphi _{n}\}$. Depending on the application these functions may be:

• A subset of the orthonormal basis of the original operator;
• A space of splines (as in the Galerkin method);
• A set of functions which approximate the eigenfunctions of the operator.

One could use the orthonormal basis generated from the eigenfunctions of the operator, which will produce diagonal approximating matrices, but in this case we would have already had to calculate the spectrum.

We now approximate $T$ by $T_{\mathcal {L}}$, which is defined as the matrix with entries

$$(T_{\mathcal {L}})_{i,j}=(T\varphi _{i},\varphi _{j}).$$

and solve the eigenvalue problem $T_{\mathcal {L}}u=\lambda u$. It can be shown that the matrix $T_{\mathcal {L}}$ is the compression of $T$ to ${\mathcal {L}}$.

For differential operators (such as Sturm-Liouville operators), the inner product $(\cdot ,\cdot )$ can be replaced by the weak formulation ${\mathcal {A}}(\cdot ,\cdot )$.

If a subset of the orthonormal basis was used to find the matrix, the eigenvectors of $T_{\mathcal {L}}$ will be linear combinations of orthonormal basis functions, and as a result they will be approximations of the eigenvectors of $T$.

Properties

Spectral pollution

It is possible for the Rayleigh–Ritz method to produce values which do not converge to actual values in the spectrum of the operator as the truncation gets large. These values are known as spectral pollution. In some cases (such as for the Schrödinger equation), there is no approximation which both includes all eigenvalues of the equation, and contains no pollution.

The spectrum of the compression (and thus pollution) is bounded by the numerical range of the operator; in many cases it is bounded by a subset of the numerical range known as the essential numerical range.

For matrix eigenvalue problems

In numerical linear algebra, the Rayleigh–Ritz method is commonly applied to approximate an eigenvalue problem

$$A\mathbf {x} =\lambda \mathbf {x}$$

for the matrix $A\in \mathbb {C} ^{N\times N}$ of size $N$ using a projected matrix of a smaller size $m<N$, generated from a given matrix $V\in \mathbb {C} ^{N\times m}$ with orthonormal columns. The matrix version of the algorithm is the most simple:

1. Compute the $m\times m$ matrix $V^{*}AV$, where $V^{*}$ denotes the complex-conjugate transpose of $V$
2. Solve the eigenvalue problem $V^{*}AV\mathbf {y} _{i}=\mu _{i}\mathbf {y} _{i}$
3. Compute the Ritz vectors ${\tilde {\mathbf {x} }}_{i}=V\mathbf {y} _{i}$ and the Ritz value ${\tilde {\lambda }}_{i}=\mu _{i}$
4. Output approximations $({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})$, called the Ritz pairs, to eigenvalues and eigenvectors of the original matrix $A$.

If the subspace with the orthonormal basis given by the columns of the matrix $V\in \mathbb {C} ^{N\times m}$ contains $k\leq m$ vectors that are close to eigenvectors of the matrix $A$, the Rayleigh–Ritz method above finds $k$ Ritz vectors that well approximate these eigenvectors. The easily computable quantity $\|A{\tilde {\mathbf {x} }}_{i}-{\tilde {\lambda }}_{i}{\tilde {\mathbf {x} }}_{i}\|$ determines the accuracy of such an approximation for every Ritz pair.

In the easiest case $m=1$, the $N\times m$ matrix $V$ turns into a unit column-vector $v$, the $m\times m$ matrix $V^{*}AV$ is a scalar that is equal to the Rayleigh quotient $\rho (v)=v^{*}Av/v^{*}v$, the only $i=1$ solution to the eigenvalue problem is $y_{i}=1$ and $\mu _{i}=\rho (v)$, and the only one Ritz vector is $v$ itself. Thus, the Rayleigh–Ritz method turns into computing of the Rayleigh quotient if $m=1$.

Another useful connection to the Rayleigh quotient is that $\mu _{i}=\rho (v_{i})$ for every Ritz pair $({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})$, allowing to derive some properties of Ritz values $\mu _{i}$ from the corresponding theory for the Rayleigh quotient. For example, if $A$ is a Hermitian matrix, its Rayleigh quotient (and thus its every Ritz value) is real and takes values within the closed interval of the smallest and largest eigenvalues of $A$.

Example

The matrix

$$A={\begin{bmatrix}2&0&0\\0&2&1\\0&1&2\end{bmatrix}}$$

has eigenvalues $1,2,3$ and the corresponding eigenvectors

$$\mathbf {x} _{\lambda =1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {x} _{\lambda =2}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {x} _{\lambda =3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.$$

Let us take

$$V={\begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix}},$$

then

$$V^{*}AV={\begin{bmatrix}2&1\\1&2\end{bmatrix}}$$

with eigenvalues $1,3$ and the corresponding eigenvectors

$$\mathbf {y} _{\mu =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {y} _{\mu =3}={\begin{bmatrix}1\\1\end{bmatrix}},$$

so that the Ritz values are $1,3$ and the Ritz vectors are

$$\mathbf {\tilde {x}} _{{\tilde {\lambda }}=1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {\tilde {x}} _{{\tilde {\lambda }}=3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.$$

We observe that each one of the Ritz vectors is exactly one of the eigenvectors of $A$ for the given $V$ as well as the Ritz values give exactly two of the three eigenvalues of $A$. A mathematical explanation for the exact approximation is based on the fact that the column space of the matrix $V$ happens to be exactly the same as the subspace spanned by the two eigenvectors $\mathbf {x} _{\lambda =1}$ and $\mathbf {x} _{\lambda =3}$ in this example.

For matrix singular value problems

Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method to find approximations to left and right singular vectors of the matrix $M\in \mathbb {C} ^{M\times N}$ of size $M\times N$ in given subspaces by turning the singular value problem into an eigenvalue problem.

Using the normal matrix

The definition of the singular value $\sigma$ and the corresponding left and right singular vectors is $Mv=\sigma u$ and $M^{*}u=\sigma v$. Having found one set (left of right) of approximate singular vectors and singular values by applying naively the Rayleigh–Ritz method to the Hermitian normal matrix $M^{*}M\in \mathbb {C} ^{N\times N}$ or $MM^{*}\in \mathbb {C} ^{M\times M}$, whichever one is smaller size, one could determine the other set of left of right singular vectors simply by dividing by the singular values, i.e., $u=Mv/\sigma$ and $v=M^{*}u/\sigma$. However, the division is unstable or fails for small or zero singular values.

An alternative approach, e.g., defining the normal matrix as $A=M^{*}M\in \mathbb {C} ^{N\times N}$ of size $N\times N$, takes advantage of the fact that for a given $N\times m$ matrix $W\in \mathbb {C} ^{N\times m}$ with orthonormal columns the eigenvalue problem of the Rayleigh–Ritz method for the $m\times m$ matrix

$$W^{*}AW=W^{*}M^{*}MW=(MW)^{*}MW$$

can be interpreted as a singular value problem for the $N\times m$ matrix $MW$. This interpretation allows simple simultaneous calculation of both left and right approximate singular vectors as follows.

1. Compute the $N\times m$ matrix $MW$.
2. Compute the thin, or economy-sized, SVD $MW=\mathbf {U} \Sigma \mathbf {V} _{h},$ with $N\times m$ matrix $\mathbf {U}$, $m\times m$ diagonal matrix $\Sigma$, and $m\times m$ matrix $\mathbf {V} _{h}$.
3. Compute the matrices of the Ritz left $U=\mathbf {U}$ and right $V_{h}=\mathbf {V} _{h}W^{*}$ singular vectors.
4. Output approximations $U,\Sigma ,V_{h}$, called the Ritz singular triplets, to selected singular values and the corresponding left and right singular vectors of the original matrix $M$ representing an approximate Truncated singular value decomposition (SVD) with left singular vectors restricted to the column-space of the matrix $W$.

The algorithm can be used as a post-processing step where the matrix $W$ is an output of an eigenvalue solver, e.g., such as LOBPCG, approximating numerically selected eigenvectors of the normal matrix $A=M^{*}M$.

Example

The matrix

$$M={\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}$$

has its normal matrix

$$A=M^{*}M={\begin{bmatrix}1&0&0&0\\0&4&0&0\\0&0&9&0\\0&0&0&16\\\end{bmatrix}},$$

singular values $1,2,3,4$ and the corresponding thin SVD

$$A={\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}{\begin{bmatrix}4&0&0&0\\0&3&0&0\\0&0&2&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}},$$

where the columns of the first multiplier from the complete set of the left singular vectors of the matrix $A$, the diagonal entries of the middle term are the singular values, and the columns of the last multiplier transposed (although the transposition does not change it)

$${\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}^{*}\quad =\quad {\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}$$

are the corresponding right singular vectors.

Let us take

$$W={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\1/{\sqrt {2}}&-1/{\sqrt {2}}\\0&0\\0&0\end{bmatrix}}$$

with the column-space that is spanned by the two exact right singular vectors

$${\begin{bmatrix}0&1\\1&0\\0&0\\0&0\end{bmatrix}}$$

corresponding to the singular values 1 and 2.

Following the algorithm step 1, we compute

$$MW={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\{\sqrt {2}}&-{\sqrt {2}}\\0&0\\0&0\end{bmatrix}},$$

and on step 2 its thin SVD $MW=\mathbf {U} {\Sigma }\mathbf {V} _{h}$ with

$$\mathbf {U} ={\begin{bmatrix}0&1\\1&0\\0&0\\0&0\\0&0\end{bmatrix}},\quad \Sigma ={\begin{bmatrix}2&0\\0&1\end{bmatrix}},\quad \mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}.$$

Thus we already obtain the singular values 2 and 1 from $\Sigma$ and from $\mathbf {U}$ the corresponding two left singular vectors $u$ as $[0,1,0,0,0]^{*}$ and $[1,0,0,0,0]^{*}$, which span the column-space of the matrix $W$, explaining why the approximations are exact for the given $W$.

Finally, step 3 computes the matrix $V_{h}=\mathbf {V} _{h}W^{*}$

$$\mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}\,{\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}&0&0\\1/{\sqrt {2}}&-1/{\sqrt {2}}&0&0\end{bmatrix}}={\begin{bmatrix}0&1&0&0\\1&0&0&0\end{bmatrix}}$$

recovering from its rows the two right singular vectors $v$ as $[0,1,0,0]^{*}$ and $[1,0,0,0]^{*}$. We validate the first vector: $Mv=\sigma u$

$${\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}$$

and $M^{*}u=\sigma v$

$${\begin{bmatrix}1&0&0&0&0\\0&2&0&0&0\\0&0&3&0&0\\0&0&0&4&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}.$$

Thus, for the given matrix $W$ with its column-space that is spanned by two exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all exactly. For an arbitrary matrix $W$, we obtain approximate singular triplets which are optimal given $W$ in the sense of optimality of the Rayleigh–Ritz method.

Applications and examples

In quantum physics

In quantum physics, where the spectrum of the Hamiltonian is the set of discrete energy levels allowed by a quantum mechanical system, the Rayleigh–Ritz method is used to approximate the energy states and wavefunctions of a complicated atomic or nuclear system.

In dynamical systems

The Koopman operator allows a finite-dimensional nonlinear system to be encoded as an infinite-dimensional linear system. In general, both of these problems are difficult to solve, but for the latter we can use the Ritz-Galerkin method to approximate a solution.

See also

• Ritz method
• Rayleigh quotient
• Arnoldi iteration

External links

• Course on Calculus of Variations, has a section on Rayleigh–Ritz method.