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ECE 595, Section 10Numerical Simulations

Lecture 9: Programming for Linear

Algebra

Prof. Peter Bermel

January 28, 2013

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Outline

Recap from Friday

Application Examples

Electrostatic potential (Poissons equation)

1D array of charge

2D grid of charge

Arrays of interacting spins

1D interaction along a chain

2D nearest-neighbor coupling

1/28/2013 ECE 595, Prof. Bermel

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Recap from Friday

Eigenproblem Solution Techniques

Power Methods

Inverse Iteration

Atomic Transformations

Factorization Methods

Linear algebra software packages

Basic Linear Algebra Subroutines (BLAS)

Linear Algebra Package (LAPACK)


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Electrostatic Potential Example

Consider an array of charges governed by

Gauss law:

=

/

Using the definition of potential yields

Poissons equation:

2

=

/

Consider solving this equation for an arbitrary

set of charges what to do?


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Electrostatic Potential Example

Strictly speaking, continuous fields have an

uncountable number of possible values, and

cannot be evaluated numerically

Key is to transform from continuous fieldvalues to those on a grid

Increase resolution as needed


1 2 3 45 6 7 8 910

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Electrostatic Potential Solution: 1D

Approximate Laplacian in 1D with:

2 1 2 + +12 Where h is the grid spacing

Sets up the linear algebra problem:2 1 01 2 10 1 2

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

2 1 0

1 2 10 1 2

123

21

=2

123

21


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In MATLAB, use:


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Use backslash operator to solve linear algebra

problems of the form =


Charge distribution

Electric potential

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Connections in two directions creates a total of 5 non-vanishingdiagonals in our linear algebra problem:

4 1 01

4 1

0 1 4

1 0 0

0 1 0

0 0 1 1 0 0

0 1 0

0 0 1

4 1 0

1 4 10 1 4

123

21

= 2

123

21


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Charge distribution in 2D

(7x7 grid)Electrostatic potential in 2D

(7x7 grid)

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Spin Array Example

Consider an array of spins , coupled by anexchange interaction

Ising-model Hamiltonian is given by:

=

+

The brackets often are interpreted to meannearest-neighbor interactions only


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Spin Array Solution: 1D

Convert Hamiltonian into matrix, assuming nearest neighborinteraction only:

=

0 0

0 0 0

0 0 0

0 0 0

0 0 00 0 0

0 0 0

0 0

Use Schrodinger equation to set up eigenproblem:

= Choose basis spinor wavefunction :

= 1, 2, ,


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In MATLAB, use the following:


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Convert Hamiltonian into matrix, assuming nearest neighborinteractions in 2 directions:

=

0 0

0 00 00 0

0 00 00 0

0 0

Use Schrodinger equation to set up eigenproblem:

= Choose basis spinor wavefunction :

= 1, 2, ,


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In MATLAB, use the following:


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Next Class

Is on Wednesday, Jan. 30

Will discuss numerical tools for

simulating eigenproblems further


2013.01.28-ece595e-l09

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