Linear boundary value problems: Matrix methods Aims: write boundary value problem for ODE as a matrix equation. use NAG (Numerical Algorithms group) routine to solve the matrix problem.

Setting up the problem: linear 2nd order ODE boundary conditions Discretize the independent variable, t.

Discretizing t: the function Continuum: Numerical:

Discretizing t, the derivatives: Taylor expand the function x(t) Solve for , Solve for , Substitute these into the continuum ODE to get a set of linear equations for the unknowns xn.

The equation: Eg. The boundary conditions: The analytic solution:

the discretized equation is Try n=1 n=2 n=3

given that the boundary conditions are: x0=0, x4=1, we arrive at which we can write in matrix form Now solve with linear algebra, e.g. row reduction, matrix inversion…

in this example we have that for general this “tri-diagonal” structure comes from the discrete version of the derivative operator. We can think of the derivative operator as connecting neighbouring lattice points, as exemplified by the above equation.

Back to Schrodinger’s equation: matrix method we want to solve subject to the boundary conditions

Back to Schrodinger’s equation: matrix method introduce the rescaled quantities introduce the finite difference derivatives to get it is our aim to find E (i.e. ) and

We then get the series of linear equations Or, in matrix form This is now a matrix eigenvalue problem, which we can solve to find the eigenvalues And the eigenvectors, n. This gives us the E we are after.

We could try to solve using the standard technique from matrix algebra; the characteristic equation This leads to a large order polynomial and is intractable analytically. Fortunately there are people who have solved this before, the Numerical Algorithms Group, NAG. We shall be using a particular routine which solves the eigenvalue, eigenvector problem for tri-diagonal matrices, “F08JEF”.

Using NAG routines in C create a folder in which your program will live write your C program, remembering to include the NAG routines, in the new folder In Plato, click on “New Project” in the file menu Click on C++ application, enter a name for your project, and choose the location to be the folder you just created. A new “Project Explorer” window will appear on the right. Right-click on “references”, then left-click on “add reference” Choose “Nag_mk21” on the P-drive, by clicking on the “MyComputer” icon Go into the “bin” folder and double-click on “FLDLL214Z_nag.dll” Right-click on “Source Files” in the “Project Explorer” window, choose “Add Existing Items” and select your C program. #include <nagmk21.h> //link to NAG library #include <stdio.h> #include <math.h> #include <stdlib.h> main(void) { … }

F08JEF: the tri-diagonal NAG routine #include <nagmk21.h> //link to NAG library #include <stdio.h> #include <math.h> #include <stdlib.h> int const N=100; //Size of matrix main(void) { char COMPZ='I'; //Calculate eigenvalues and eigenvectors int CLEN=1; //Character length int INFO=0; //Gives error information double D[N]; //Input diagonal elements //and returns eigenvalues double E[N-1]; //Input off diagonal elements double Z[N][N]; //Returns array of eigenvectors //as Z[quantum state][i] double WORK[2*(N-1)]; //Work space int i; for(i=0;i<N;i++) D[i]= 1.0; //Diagonal elements go here for(i=0;i<N-1;i++) E[i]= 2.0; //Off diagonal elements go here F08JEF(&COMPZ,CLEN,&N,(double *)D,(double *)E,(double *)Z,&N,(double *)WORK,&INFO); printf("%f\n",D[i]); //Print eigenvalues printf("%f\n",Z[0][i]); //Print first eigenvector }

F08JEF and the harmonic oscillator: For the harmonic oscillator the diagonal elements are the off-diagonal elements are E[i]=-1; the results depend on N and x N=500, x=0.1 gives En=0.994.., 2.9969.., 4.9919.. N=100, x=0.1 gives En=0.994.., 2.9969.., 4.9919.. N=50, x=0.1 gives En=1.007.., 3.083.., 5.397.. the higher eigenvalues are not so well measured because the eigenfunctions spread out beyond our lattice size.