MATH 685/CSI 700 Lecture Notes

Lecture 1.

Intro to Scientific Computing

Useful info Course website:

http://math.gmu.edu/~memelian/teaching/Spring10 MATLAB instructions:

http://math.gmu.edu/introtomatlab.htm Mathworks, the creator of MATLAB: http://www.mathworks.com OCTAVE = free MATLAB clone

Available for download at http://octave.sourceforge.net/

Scientific computing Design and analysis of algorithms for numerically solving

mathematical problems in science and engineering

Deals with continuous quantities vs. discrete (as, say, computer science)

Considers the effect of approximations and performs error analysis

Is ubiquitous in modern simulations and algorithms modeling natural phenomena and in engineering applications

Closely related to numerical analysis

Develop mathematical model (usually requires a combination of math skills and some a priori knowledge of the system)

Come up with numerical algorithm (numerical analysis skills)

Implement the algorithm (software skills)

Run, debug, test the software

Visualize the results

Interpret and validate the results

Computational problems:attack strategy

Mathematical modeling

Computational problems:well-posedness The problem is well-posed, if

(a) solution exists

(b) it is unique

(c) it depends continuously on problem data

The problem can be well-posed, but still sensitive to perturbations. The algorithm should attempt to simplify the problem, but not make sensitivity worse than it already is.

Simplification strategies:

Infinite finite

Nonlinear linear

High-order low-orderOnly

approximate

solutio

n can be

obtained this

way!

Sources of numerical errors Before computation

modeling approximations empirical measurements, human errors previous computations

During computation truncation or discretization Rounding errors

Accuracy depends on both, but we can only control the second part Uncertainty in input may be amplified by problem Perturbations during computation may be amplified by algorithm

Abs_error = approx_value – true_valueRel_error = abs_error/true_value

Approx_value = (true_value)x(1+rel_error)

Can be controlled througherror analysis

Cannot be controlled

Sources of numerical errors Propagated vs. computational error

x = exact value, y = approx. value F = exact function, G = its approximation G(y) – F(x) = [G(y) - F(y)] + [F(y) - F(x)]

Rounding vs. truncation error Rounding error: introduced by finite precision calculations in the

computer arithmetic Truncation error: introduced by algorithm via problem simplification, e.g.

series truncation, iterative process truncation etc.

Total error = Computational error:affected by algorithm

Propagated data error:not affected by algorithm+

Computational error = Truncation error + rounding error

Backward vs. forward errors

Backward error analysis How much must original problem change to give result actually

obtained?

How much data error in input would explain all error in computed result?

Approximate solution is good if it is the exact solution to a nearby problem

Backward error is often easier to estimate than forward error

Backward vs. forward errors

Conditioning Well-conditioned (insensitive) problem: small relative change in

input gives commensurate relative change in the solution Ill-conditioned (sensitive): relative change in output is much larger

than that in the input data Condition number = measure of sensitivity

Condition number = |rel. forward error| / |rel. backward error|

= amplification factor

Conditioning

Stability Algorithm is stable if result produced is relatively insensitive to

perturbations during computation

Stability of algorithms is analogous to conditioning of problems

From point of view of backward error analysis, algorithm is stable if result produced is exact solution to nearby problem

For stable algorithm, effect of computational error is no worse than effect of small data error in input

Accuracy Accuracy : closeness of computed solution to true solution of problem

Stability alone does not guarantee accurate results

Accuracy depends on conditioning of problem as well as stability of algorithm

Inaccuracy can result from applying stable algorithm to ill-conditioned problem or unstable algorithm to well-conditioned

problem

Applying stable algorithm to well-conditioned problem yields accurate solution

Floating point representation

Floating point systems

Normalized representation

Not all numbers can be represented this way, those that can are called machine numbers

Rounding rules If real number x is not exactly representable, then it is approximated by

“nearby” floating-point number fl(x)

This process is called rounding, and error introduced is called rounding error

Two commonly used rounding rules chop: truncate base- expansion of x after (p − 1)st digit; also called round toward zero round to nearest : fl(x) is nearest floating-point number to x, using floating-

point number whose last stored digit is even in case of tie; also called round to even

Round to nearest is most accurate, and is default rounding rule in IEEE systems

Floating point arithmetic

Machine precision

Floating point operations

Summing series in floating-point arithmetic

Loss of significance

Loss of significance

Loss of significance

Loss of significance