A method for solving problems in
variational calculus
and, in general, finite-dimensional extremal problems, based on
optimization of a functional on finite-dimensional subspaces or manifolds.
Let the problem of finding a minimum point of a functional
on a separable
Banach space
be posed, where
is bounded from below. Let some system of elements
,
complete in
(cf.
Complete system),
be given (a so-called coordinate system). In the
Ritz method, the minimizing element in the
-th
approximation is sought in the linear hull of the first
coordinate elements
,
i.e. the coefficients
of the approximation
are defined by the condition that
be minimal among the specified elements. Instead of a
coordinate system one can specify a sequence of subspaces
,
not necessarily nested.
Let
be a Hilbert space with scalar product
,
let
be a self-adjoint positive-definite (i.e.
:
for all
),
possibly unbounded, operator in
,
and let
be the Hilbert space obtained by completing the domain of definition
of
with respect to the norm
generated by the scalar product
,
.
Let it be required to solve the problem
This is equivalent to the problem of finding a minimum point of the quadratic functional
which can be written in the form
where
is a solution of equation
(1).
Let
,
be closed (usually, finite-dimensional) subspaces such that
as
for every
,
where
is the orthogonal projection in
projecting onto
.
By minimizing
in
one obtains a
Ritz approximation
to the solution of equation
(1);
moreover,
as
.
If
and
is a basis in
,
then the coefficients of the element
are determined from the linear system of equations
One can also arrive at a Ritz approximation without making use of
the variational statement of the problem
(1).
Namely,
by defining the approximation
(2)
from the condition
(the
Galerkin method),
one arrives at the same system of equations
(3).
That
is why the Ritz method for equation
(1)
is sometimes called the
Ritz–Galerkin method.
Ritz's method is widely applied when solving eigenvalue problems,
boundary value problems and operator equations in general. Let
and
be self-adjoint operators in
.
Moreover, let
be positive definite,
be positive,
,
and let the operator
be completely continuous in
(cf.
Completely-continuous operator).
By virtue of the above requirements,
is self-adjoint and positive in
,
and the spectrum of the problem
consists of positive eigenvalues:
Ritz's method is based on a variational determination of eigenvalues. For instance,
by carrying out minimization only over the subspace
one obtains Ritz approximations
of
.
If
is, as above, a basis in
,
then the Ritz approximations
of
,
,
are determined from the equation
and the vector of coefficients
of the approximation
to
is determined as a non-trivial solution of the linear homogeneous system
.
The Ritz method provides an approximation from above of the eigenvalues, i.e.
,
.
If the
-th
eigenvalue of problem
(4)
is simple
,
then the convergence rate of the Ritz method is characterized by the following relations:
where
as
.
Similar relations can be carried over to the case of multiple
,
but then they need certain refinements (see
[2]).
W. Ritz
[4]
proposed his method in
1908,
but even earlier
Lord Rayleigh
had applied this method to solve certain eigenvalue problems. In
this connection the Ritz method is often called the
Rayleigh–Ritz method,
especially if one speaks about solving an eigenvalue problem.