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UA / Mathematics / MATH 511 / What are the competing methods in numerical analysis?

What are the competing methods in numerical analysis?

What are the competing methods in numerical analysis?

Description

School: University of Alabama - Tuscaloosa
Department: Mathematics
Course: Numerical Analysis I
Professor: Roger sidje
Term: Spring 2016
Tags: Math, Numerical Analysis, MATH 300, and Sidje
Cost: Free
Name: UA Math 300 Week 1 & 2 Notes
Description: This set of notes covers the classes 1/14/16, 1/19/16, and 1/21/16. These are my version of notes for Math 300 with Dr. Sidje. This first week (which covers week 1 and 2!) are free!
Uploaded: 01/23/2016
13 Pages 231 Views 3 Unlocks
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1/14/16 Introduction To Numerical Analysis

  • Design / implement                 algorithms / methods
  • Analyze when / why they work / fail
  • Classify competing methods

Don't forget about the age old question of What question is being answered by Epistomology?

Example 1:

        Find the area under the unit circle shifted up by 1. (in the first quadrant)

We also discuss several other topics like a hydrated lithium iodide sample, lii·xh2o, weighing 4.450 g is dried in an oven. when the anhydrous salt is removed from the oven, its mass is 3.170 g. what is the value of x?

Equation of circle: x2 + y2 = r2 = 1

        Solve for y → y = If you want to learn more check out What is the distance light travels in one year?

When you shift it by 1:        y =

  1. Geometrically: S = (area of unit circle) + area of unit square

                = (1)2 + 1 = 1.7854

  1. Analytically (by a closed - form formula)

        S = f(x)dx =  =  + arcsin x + xIf you want to learn more check out How does our nervous system work?

Evaluate in matlab

  • Define a function f(x) =  +  arcsin x + x by using F = @(x) (x/2)*sqr + (1 - x2) - (½) arcsin x + x

Then to evaluateDon't forget about the age old question of f 288 scantron

        F(1) - F(0)

        Answer = 1.7854

  1. Riemann Sum

We also discuss several other topics like What refers to the study of microscopic structures and tissues?

A = A1 + A2 + A3 + A4

= f(x1)x + f(x2)x + f(x3)x + f(x4)

f(xi)

Riemann Sums continued → in Matlab

a =0 b =1 n = 4 (all separate names)

Dx = (b - a)/n

        Dx = ¼  = 2500

→ x = [1.n] x dx

        X = 0.2500        0.5000         0.7500        1.0000

Format rat

X

X = ¼        ½ ¾ 1

F = @(x) sqrt(1 - x2) + 1

A = sum (fx)) . dx

A = 3217 / 1981

Format  

A = 1.6239

Then you can easily redo everything for a larger n, thus a closer approximate

Say you use n = 10,000

Dx =

X =

A =

A = 1.7853 → off by only .0001

Method 3 using Matlab is the best for complicate, real - life problems

Example 2: Find ln(2)

  1. Riemann Sum

Lnx = dt

Ln2 = dx

F = @(x) 1. /x                                x1 = a + ix

Dx = (b.a) / n                                i = 0 = n

X = a + [1:n] = dx

A = sum(f(x) x dx)

A = .9950

H = 1000000

X = a + [1:n] x dx

A = sum(f(x)x dx)

A = .6931

log(2) = .6931

  1. Series - ln(1 - x) = ∫dx = ∫(1 + x + x2 +,...)dx

 = x + x+ … then x = -1        -ln(2) = -(1 ---)

Note: Review of Geometric series

        Sn - xSn = 1 = xn+1

        (1 - x) Sn = 1 - xn+1 → Sn =

         = x < 1 →  =         x < 1

                = 1 + x + x2

Note - he used octet, after done of Matlab

Example: Evaluate ln(2)

**Riemann Sum

        ln(x) = ∫1x dt

        ln(2) = ∫12 dt

A = f(x1)        ==← with right endpoint

                        X1 = a + ix

A = f(xi-1) aka. For n = 1 you use X0,etc ← with left endpoints

Or        A = f(xi)

Octave 1 > a = 1

Octave 2 > b = 2

Octave 3 > n = 10,dx = (b-a)/n

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