In the previous post I talked about one of the less quoted results in the variational theory of Strurm-Liouville eigenvalue problems which I owe to the book on differential equations by Erich Kamke and which I reworked for the purposes of my dissertation.

The other result that I used to complement the compulsory discussion of the Ritz method which I also found in Kamke's book states an alternative variational principle which can be used to obtain estimates of the eigenvalues. This method gives a weaker upper bound than the classical Rayleigh quotient, however takes less computational effort to evaluate. In return, one can widen the class of trial functions without severely complicating the calculations.

The expression of this alternative principle took some time to sink in, especially given the fact that I was running out of time to complete the thesis. I nearly posted a question on MSE to help me work out the motivation behind the new quotient. I am glad I did not do it, as the derivation turned out to be much easier then it seemed.

We start off with the Rayleigh quotient written in the abstract form:

$$J\left\{ \phi\right\}=\frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }$$

Then using Cauchy-Schwarz inequality we obtain.

$$\left(\left\langle \phi,L\left[\phi\right]\right\rangle \right)^{2}\leq\left\langle \phi,\phi\right\rangle \left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle

\frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }\leq\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }$$

Written explicitly

$$\frac{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}{\int_{\Omega}\rho\phi^{2}d\boldsymbol{x}}\le\frac{\int_{\Omega}\rho\left(L\left[\phi\right]\right)^{2}d\boldsymbol{x}}{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}$$

Hence we can replace the original problem with the following one

$$K\left\{ \phi\right\} =\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }\to\min,\qquad\phi\in\mathfrak{A}$$

Consider the following eigenvalue problem

$$-y''=\lambda y,\quad y\left(0\right)=y\left(1\right)=0$$

Take the following trial function

$$y_{1}=\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}$$

First we perform the calculation using the Rayleigh quotient

$$J\left\{ y_{1}\right\} =\frac{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},y_{1}\right\rangle }=-\frac{\int_{0}^{1}y_{1}y_{1}''dx}{\int_{0}^{1}y_{1}^{2}dx}$$

$$\begin{aligned}\int_{0}^{1}y_{1}y_{1}''dx & =\int_{0}^{1}\left[\left(\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}\right)\left(-x+x^{2}\right)\right]dx\\

& =\int_{0}^{1}\left(\frac{x^{6}}{12}-\frac{x^{5}}{4}+\frac{x^{4}}{6}+\frac{x^{3}}{12}-\frac{x^{2}}{12}\right)dx\\

& =-\frac{17}{5040}

\end{aligned}$$

The factor just evaluated will be common for both variational quotients

$$\begin{aligned}\int_{0}^{1}y_{1}^{2}dx & =\int_{0}^{1}\left(\frac{x^{8}}{144}-\frac{x^{7}}{36}+\frac{x^{6}}{36}+\frac{x^{5}}{72}+\frac{x^{2}}{144}\right)dx\\

& =\frac{31}{90720}

\end{aligned}$$

$$ \lambda_{1}\le J\left\{ y_{1}\right\} =\frac{17\cdot90720}{5040\cdot31}\approx9.871$$

Now we use the alternative variational principle

$$K\left\{ y_{1}\right\} =\frac{\left\langle L\left[y_{1}\right],L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }=-\frac{\int_{0}^{1}y_{1}''^{2}dx}{\int_{0}^{1}y_{1}y_{1}''dx}$$

Obviously the gain in computational efficiency comes from replacing $y^{2}$ with $\left(L\left[y\right]\right)^{2}$ which will be a polynomial of the order less by 4, than $y^{2}$.

$$\int_{0}^{1}y_{1}''^{2}dx=\int_{0}^{1}\left(x^{4}-2x^{3}+x^{2}\right)dx=\frac{1}{30}$$

$$\lambda_{1}\le K\left\{ y_{1}\right\} =\frac{5040}{17\cdot30}\approx9.882$$

Hence

$$K\left\{ y_{1}\right\} >J\left\{ y_{1}\right\} >\lambda_{1}=\pi^{2}$$

as expected, however $K\left\{ y_{1}\right\}$ takes less operations to evaluate.

The other result that I used to complement the compulsory discussion of the Ritz method which I also found in Kamke's book states an alternative variational principle which can be used to obtain estimates of the eigenvalues. This method gives a weaker upper bound than the classical Rayleigh quotient, however takes less computational effort to evaluate. In return, one can widen the class of trial functions without severely complicating the calculations.

The expression of this alternative principle took some time to sink in, especially given the fact that I was running out of time to complete the thesis. I nearly posted a question on MSE to help me work out the motivation behind the new quotient. I am glad I did not do it, as the derivation turned out to be much easier then it seemed.

We start off with the Rayleigh quotient written in the abstract form:

$$J\left\{ \phi\right\}=\frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }$$

Then using Cauchy-Schwarz inequality we obtain.

$$\left(\left\langle \phi,L\left[\phi\right]\right\rangle \right)^{2}\leq\left\langle \phi,\phi\right\rangle \left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle

\frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }\leq\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }$$

Written explicitly

$$\frac{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}{\int_{\Omega}\rho\phi^{2}d\boldsymbol{x}}\le\frac{\int_{\Omega}\rho\left(L\left[\phi\right]\right)^{2}d\boldsymbol{x}}{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}$$

Hence we can replace the original problem with the following one

$$K\left\{ \phi\right\} =\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }\to\min,\qquad\phi\in\mathfrak{A}$$

**Example**.Consider the following eigenvalue problem

$$-y''=\lambda y,\quad y\left(0\right)=y\left(1\right)=0$$

Take the following trial function

$$y_{1}=\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}$$

First we perform the calculation using the Rayleigh quotient

$$J\left\{ y_{1}\right\} =\frac{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},y_{1}\right\rangle }=-\frac{\int_{0}^{1}y_{1}y_{1}''dx}{\int_{0}^{1}y_{1}^{2}dx}$$

$$\begin{aligned}\int_{0}^{1}y_{1}y_{1}''dx & =\int_{0}^{1}\left[\left(\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}\right)\left(-x+x^{2}\right)\right]dx\\

& =\int_{0}^{1}\left(\frac{x^{6}}{12}-\frac{x^{5}}{4}+\frac{x^{4}}{6}+\frac{x^{3}}{12}-\frac{x^{2}}{12}\right)dx\\

& =-\frac{17}{5040}

\end{aligned}$$

The factor just evaluated will be common for both variational quotients

$$\begin{aligned}\int_{0}^{1}y_{1}^{2}dx & =\int_{0}^{1}\left(\frac{x^{8}}{144}-\frac{x^{7}}{36}+\frac{x^{6}}{36}+\frac{x^{5}}{72}+\frac{x^{2}}{144}\right)dx\\

& =\frac{31}{90720}

\end{aligned}$$

$$ \lambda_{1}\le J\left\{ y_{1}\right\} =\frac{17\cdot90720}{5040\cdot31}\approx9.871$$

Now we use the alternative variational principle

$$K\left\{ y_{1}\right\} =\frac{\left\langle L\left[y_{1}\right],L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }=-\frac{\int_{0}^{1}y_{1}''^{2}dx}{\int_{0}^{1}y_{1}y_{1}''dx}$$

Obviously the gain in computational efficiency comes from replacing $y^{2}$ with $\left(L\left[y\right]\right)^{2}$ which will be a polynomial of the order less by 4, than $y^{2}$.

$$\int_{0}^{1}y_{1}''^{2}dx=\int_{0}^{1}\left(x^{4}-2x^{3}+x^{2}\right)dx=\frac{1}{30}$$

$$\lambda_{1}\le K\left\{ y_{1}\right\} =\frac{5040}{17\cdot30}\approx9.882$$

Hence

$$K\left\{ y_{1}\right\} >J\left\{ y_{1}\right\} >\lambda_{1}=\pi^{2}$$

as expected, however $K\left\{ y_{1}\right\}$ takes less operations to evaluate.