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Lecture 19, Review 2 


Go over WEB pages Lecture 11 through 18.
Open book, open notes, no study guide or copy.
Read the instructions.
Follow the instructions, else lose points.
Read the question carefully.
Answer the question that is asked, not the question you want to answer.


Some things you should know:

The roots of a polynomial with real coefficients can be complex.

Some languages have "complex" as a built in type (e.g. Fortran).
Many languages have complex arithmetic available: Ada, Fortran,
MatLab, Mathematica, Python.

Some languages provide standard packages for "complex" (e.g. Ada and C++)

Some languages require you find or make your own "complex" (e.g. C)

All elementary functions are defined and computable for complex arguments.

All complex elementary functions may be computed with only real functions.

Example: sin(z)=sin(re z)*cosh(im z) + i cos(re z)*sinh(im z)

Many languages have either built in or available complex elementary
functions, e.g. Ada, C++, Fortran, Matlab, Mathematica, Java, Python
and some others.

Some complex elementary functions are difficult to compute accurately. 

Some elementary functions have a branch cut in the complex plane.

The branch cut is often along the negative real axis.

The derivative is not defined across the branch cut.

Eigenvalues must satisfy several definitions.

There are exactly n eigenvalues for every n by n matrix.

The eigenvalues of a matrix of real elements may be complex.

The determinant of a matrix with a variable subtracted from each
diagonal element yields the characteristic equation of the matrix.
e.g  det|A - eI| = c_n*e^n + ... _ c_1*e + c_0

The roots of the characteristic equation are eigenvalues.

det|A - eI|=0 is one definition of eigenvalues.

Given a polynomial equation, the coefficients of the polynomial may
be placed in a matrix such that the eigenvalues of the matrix are
the roots of the polynomial. 

An eigenvector exists for every eigenvalue. Av = ev

It is always possible to find a set of orthogonal eigenvectors
that form the basis of an n dimensional space for an n by n matrix.

Eigenvectors may be complex when the matrix elements and the
eigenvalues are real.

The eigenvectors of a matrix of all ones are usually chosen as the
unit vectors [1 0 ... 0] [0 1 0 ... 0] ... [0 ... 0 1]

The equation  A v = e v must be satisfied for every eigenvalue, e,
and its corresponding eigenvector, v. |v| non zero.

There are many possible test for a program that is supposed to compute
eigenvalues and eigenvectors.

MatLab may not produce unique eigenvectors using [v e]=eig(a)

MatLab may not produce eigenvectors that are mutually orthogonal.

LAPACK is a source of many numerical algorithms implemented in Fortran.

LAPACK code may be converted to any other language.

LAPACK object files may be used with most languages via some interface.

LAPACK source code needs the BLAS, Basic Linear Algebra Subroutines.

The BLAS may be available, optimized, for your computer.

LAPACK is available with files to compile and install on
many computers from  www.netlib.org/lapack

TOMS is numerical source code from the ACM Transactions on Mathematical
software and may be found on www.netlib.org/toms
 
Some problems require more than double precision floating point.

"gmp" is the gnu multiple precision library of "C" functions.

gmp is available for free download from the Internet.

gmp provides multiple precision integer, rational and floating point.

Any language can implement multi-precision, Java has it available,
other languages use libraries.

52! has too many significant digits to exactly represent in double
precision floating point.

Solving large systems of equation may require multi-precision in order
to get reasonable results.

A simple example of the need for complex roots is x^2+1 = 0
with roots i and -i .

The roots of high order polynomials may be difficult to compute
accurately, e.g. x^16 + 5 = 0 .

There is no algorithm that can guarantee finding all the
roots of every polynomial to a specified accuracy.

Newtons method works most of the time for finding complex roots.
z_next = z - P(z)/P'(z)
(Using a different initial guess usually fixes the problem.)

Optimization is finding the values of independent variables such
that a function is minimized.

Optimizing the absolute value of a function finds the values of
the independent variables that make the function closest to zero.

There is no algorithm that can guarantee finding the optimum
value of every function to a required accuracy.

There are many methods of implementing code to find local minima.
The code varies depending on the assumption of a differentiable
function and on the assumption of no local minima.

The spiral trough is one test case that can be used to test
optimization code.

A Fourier transform of a finite set of data assumes that the
data is repeated infinitely many times before and after the data.

A FFT, Fast Fourier Transform implements a discrete Fourier transform
using order n log n computations for an n point transform. Typically
n must be a power of 2.

The FFT of real data may be complex.

The inverse FFT of the result of and FFT will be reasonably close
to the original data.

The Discrete Fourier transform and FFT of data sampled at time
steps dt will be the frequency spectrum with highest unaliased
frequency 1/(2*dt), the Nyquist frequency.

Digital Filters can be low-pass, high-pass, or band-pass.
A band-pass filter requires two frequencies.
The cutoff frequency is typically 3db down (half power) and
the amplitude decreases faster for higher order filters.

Digital Filters can be described by a set of "a" coefficients
for the numerator and "b" coefficients for the denominator.

Sound power is measured in db, decibel. 0 db for minimum
threshold of hearing. db increases as 10 log10(new_power/old_power)
60 db is 100 times the power of 40 db.

Sound can hurt, 130 db, and greater values may damage buildings.


New applications are happening. Not on the exam,
yet may be of interest:
FFT used on light


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