ME 2016 Exam 1

Numerical Methods

computing techniques that help us approximate solutions to mathematical problems

Why do we use numerical methods?

Many engineering problems have no analytical solution, require analytical solutions that are intractable or impractical, and are limited by discrete data

What kinds of engineering problems are we interested in solving via numerical methods?

Engineering Analysis and Engineering design

Engineering Analysis

displacements, stresses in a physical system

Engineerign design

size, shape, and material properties for a given physical system to support its function

A mathematical model expresses

essential features of a physical system or process in mathematical terms

Components of a mathematical model

independent variable, dependent variable, parameters, forcing function

computer program

a finite set of instructions in a programming language for a computer to execute

Algorithim

a finite set of instructions used to solve a specific problem or perform a computation

Modular Programming

dividing a computer program into small sub-programs

Characteristics of module

independent and self-contained, performs a specific will-defined operation, has one entry and exit point

Why modular programming?

writing code for one operation allows reuse, facilitates modification to the operation, makes programming easier, facilitates use of code by others

accuracy

how closely the computed or measured value agrees with the true value

Precision

how closely individual computed or measured values agree with each other

inaccuracy

a systematic deviation from truth(bias)

inprecision

magnitude of scatter(uncertainty)

True error, Et, formula

true value - approximation

True relative error, et, formula

true value - approximation/ true value * 100

True relative error

takes into account order of magnitude

approximate relative error, ea, formula

current approximation - previous approximation/ current approximation * 100

Approximate relative error is based on

iterative numerical approach where approximations get closer to true value

Types of non-numerical error: Blunders

computer or human mistakes(not common today)

Types of non-numerical error: Formulation/Model Errors

due to use of poorly concieved mathematical model(does not accurately represent the physics)

Types of non-numerical error: Data Uncertainty

Due to imprecise or inaccurate measurements

Types of numerical error: Truncation Error

results when approximations are used to represent exact mathematical procedure

Types of numerical error: Roundoff Error

results when numbers having limited significant figures are used to represent exact numbers

Truncation error example

finite divided difference approximation of derivative

Round error example

pi is about 3.14

Why are significant figures omitted in computing techniques

computers can only retain a fixed number of digits and base 2 representation cannot properly represent certain base 10 numbers exactly

floating point representation

enables us to represent large numbers and fractions in the computer

Floating point formula

fl(x) = (-1)^s (1.f) * (2^b-bias)

floating point mantissa

finite set of digits from 0<= di <= beta-1

How many bits in floating point mantissa

t

In base 2 d0=1 so we get

t+1 digits by storing only t

In IEEE754 what is s

1 bit sign

In IEEE754 what is b

k bits exponent

In IEEE754 what is f

t bits for the fraction portion of the mantissa

exponent for IEEE754

b-bias

IEEE754 bias formula

2^(k-1) - 1

denormalized numbers

enables us to represent 0 and enables gradual underflow

underflow

error that occurs when you attempt to access numbers between 0 and the first positive normalized number

overflow

error when you attempt to access numbers larger than the largest normalized number

When the exponent and fraction is all ones f=111….11 we have

NaN

Single precision

s=1, k=8, t=23

Double precision

s=1, k=11, t=52

Lower endpoint formula

bmin+1 - bias

Upper endpoint formula

bmax-1 - bias

machine precision

difference between 1 and the next largest floating point number

machine precision formula

e = beta^-t

An nth-order Taylor polynomial will be exact

for a nth order polynomial

Using Taylor series for functions such as sin,cos,exp

will not be exact

In Taylor series polynomial approxination the remainder term captures

the error

In Taylor series approximation truncations error is proportional to

the order of h^n+1

O(h) when you half stepsize will

half error

O(h²) when you half stepsize will

quater error

If stepsize h is sufficiently small

the first and lower order terms account for most of error

how does rounding error change as step size decreases

increases

how does truncation error change as step size decreases

decreases

How do we know at least one root exists

if f(x) is real and continuous on the interval xl to xu and f(xl) and f(xu) have oppisite signs

Bracketing Methods(Bisection and False Position) are always

convergent

Why might false position be slower than bisection even thought that not usually the case

tends to be one-sided since one of the points remains fixed which can sometimes lead to slow convergence

Why might false position be more efficient the bisection method

it considerers the magnitude of f(xu) and f(xl) in bracketing

When are incremental search methods convergent

monotonic functions with a single root

bracket methods require

2 initial guesses

open methods are root finding methods that

do not require starting values to bracket the root

Open methods may

diverge and typically converge faster than bracketing

Fixed Point Theorem

g(h) is continuous in an interval, [a, b], and a ≤ g(h) ≤ b (existence) and the fixed point is unique if g’ (h) exists and |g′ (ℎ)| ≤ 𝜌 < 1 for all h ∈ [a, b] (uniqueness)

Fixed point error is

linearly convergent

pitfalls of newton-raphson

derivative difficult or expensive to calculate and may diverge, oscillate, or jump roots

Newton-Raphson: oscillations may persist around

local min or max

Newton-Raphson: Initial guess near 1 root can

jump to location several roots away

Newton-Raphson: zero slope leads to

divergence and divide by zero error

When a function does not change sign at a multiple root

bracketing methods fail

when the derivative is 0 at a multiple root

newton-Raphson and secant experience divide by 0 error

Newton-Raphson and secant error for a multiple root function is

linearly convergent

curve fitting

techniques to fit curves to discrete data values to estimate values in between or simplify a complicated function

regression

curve is designed to follow the pattern of the data without attempting to intersect each point

interpolation

curve is designed to pass through each data point

trend analysis

process of using pattern of data to make predictions

hypothesis testing

an existing mathematical model is compared to measured data

residual

discrepancy between the true measured values and the approximate model values predicted by regression

least squares regression

minimize the sum of squares of residuals for all available data

coefficient of determination

quantifies the improvement of error reduction due to describing the data with least squares regression rather than average value

Describe in words what the bias is used for in IEEE standard of floating-point representation

The exponent is stored as an unsigned integer and the bias allows us to achieve positive and negative values of the exponent

In the IEEE754 standard for floating point representation, if the mantissa can store t bits, why do we get t+1?

When we normalize in base-2 the first digit is always 1 so we don’t need to store it

What do the normal equations represent

simultaneous equations that can be solved to find best fit coefficients, necessary conditions to minimize Sr, derivatives of Sr with respect to coefficients set equal to 0

IEEE754 an exponent value of all 1’s (11111) and a mantissa of all zeros represents

Infinity