1. 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.

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

3. Discretizing t: the function
Continuum:
Numerical:

4. 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.

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

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

7. 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…

8. 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.

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

10. 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

11. 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.

12. 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”.

13. 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) { … }

14. 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 }

15. 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= , ,
N=100, x=0.1 gives En= , ,
N=50, x=0.1 gives En= , ,
the higher eigenvalues are not so well measured because the eigenfunctions spread out beyond our lattice size.