Linear boundary value problems:

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.

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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.

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The equation
Eg.
The boundary conditions
The analytic solution
6

• the discretized equation is

Try

• n1
• n2
• n3

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• given that the boundary conditions are x00,
  x41, 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.

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Back to Schrodingers equation matrix method

• we want to solve
• subject to the boundary conditions

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Back to Schrodingers equation matrix method

• introduce the rescaled quantities
• introduce the finite difference derivatives to
  get
• it is our aim to find E (i.e. ?) and

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• 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.
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• 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.

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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 ltnagmk21.hgt //link to NAG
library include ltstdio.hgt include
ltmath.hgt include ltstdlib.hgt main(void)
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F08JEF the tri-diagonal NAG routine
include ltnagmk21.hgt //link to NAG
library include ltstdio.hgt include
ltmath.hgt include ltstdlib.hgt int const N100
//Size of matrix main(void) char
COMPZ'I' //Calculate eigenvalues and
eigenvectors int CLEN1
//Character length int INFO0
//Gives error information double DN
//Input diagonal elements
//and returns eigenvalues double
EN-1 //Input off diagonal elements
double ZNN //Returns array of
eigenvectors //as
Zquantum statei double WORK2(N-1)
//Work space int i for(i0iltNi)
Di 1.0 //Diagonal elements go here
for(i0iltN-1i) Ei 2.0
//Off diagonal elements go here F08JEF(COMPZ,CLEN
,N,(double )D,(double )E,(double
)Z,N,(double )WORK,INFO) for(i0iltNi)
printf("f\n",Di) //Print eigenvalues
for(i0iltNi) printf("f\n",Z0i)
//Print first eigenvector
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F08JEF and the harmonic oscillator

• For the harmonic oscillator the diagonal
  elements are
• the off-diagonal elements are
• Ei-1

• the results depend on N and ?x
• N500, ?x0.1 gives En0.994.., 2.9969..,
  4.9919..
• N100, ?x0.1 gives En0.994.., 2.9969..,
  4.9919..
• N50, ?x0.1 gives En1.007.., 3.083..,
  5.397..

• the higher eigenvalues are not so well measured
  because the eigenfunctions spread out beyond our
  lattice size.