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Numerical Methods for Engineers13.002

Introduction to Numerical Analysis fo

Fundamentals of Digital Compu Digital Computer Models

Convergence, accuracy and stabilit

Number representation

Arithmetic operations Recursion algorithms

Error Analysis Error propagation numerical stab

Error estimation

Error cancellation Condition numbers

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Digital Computer Models

x

w(x,t)

xw(x,t)

n n

xn

m

Continuous Model

Discrete Model

Different

Differenc

System o

Linear Syste

Eigenval

Non-triv

Accuracy and Sta

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m M

b Be E

Floating Number Representati

Examples

Convention

Decimal

Binary

Decimal

Binary

General

Max mantissa

Min mantissa

Max exponent

Min exponent

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Arithmetic Operations

Number Representation

Absolute Error

Relative Error

Addition

Multiplic

R

A

R

Shift mantis

Result has ex

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Recursion

Numerically evaluate square-root

Initial guess

Test

Mean of guess and its reciprocal

Recursion Algorithm

a=26;

n=10;

g=1;

sq(1)=g;for i=2:n

sq(i)= 0.5*(

end

hold off

plot([0 n],[s

hold on

plot(sq,'r')

plot(a./sq,'r

plot((sq-sqrt

grid on

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Recursion

Horners Scheme

% Horners

% for eva

a=[ 1 2 3

n=length

z=1;

b=a(1);

% Note in

for i=1:n

b=a(i

end

p=b

Evaluate polynomial

Horners Scheme

General order n

Recurrence relation

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Recursion

Order of Operations Matter

0 1

N=20; sum=0; sumr=0;

b=1; c=1; x=0.5;

xn=1;

% Number of significant dig=2;

ndiv=10;

for i=1:N

a1=sin(pi/2-pi/(ndiv*i))

a2=-cos(pi/(ndiv*(i+1)))

% Full matlab precision

xn=xn*x;

addr=xn+b*a1;

addr=addr+c*a2;

ar(i)=addr;

sumr=sumr+addr;

z(i)=sumr;

% additions with dig sig

add=radd(xn,b*a1,dig);

add=radd(add,c*a2,dig);

% add=radd(b*a1,c*a2,dig

% add=radd(add,xn,dig);

a(i)=add;

sum=radd(sum,add,dig);

y(i)=sum;

end

sumr

' i delta

res=[[1:1:N]' ar' z' a'

hold off

a=plot(y,'b'); set(a,'Li

hold on

a=plot(z,'r'); set(a,'Li

a=plot(abs(z-y)./z,'g');

legend([ num2str(dig) '

Result of small, but significantterm destroyed by subsequent

addition and subtraction of almost

equal, large numbers.

Remedy:

Change order of additions

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recur.m

>> recur

b = 1; c = 1; x = 0.5;

dig=2

i delta Sum delta(approx) Sum(approx)

res =

1.0000 0.4634 0.4634 0.5000 0.5000

2.0000 0.2432 0.7065 0.2000 0.7000

3.0000 0.1226 0.8291 0.1000 0.8000

4.0000 0.0614 0.8905 0.1000 0.9000

5.0000 0.0306 0.9212 0 0.90006.0000 0.0153 0.9364 0 0.9000

7.0000 0.0076 0.9440 0 0.9000

8.0000 0.0037 0.9478 0 0.9000

9.0000 0.0018 0.9496 0 0.9000

10.0000 0.0009 0.9505 0 0.9000

11.0000 0.0004 0.9509 0 0.9000

12.0000 0.0002 0.9511 0 0.9000

13.0000 0.0001 0.9512 0 0.9000

14.0000 0.0000 0.9512 0 0.9000

15.0000 0.0000 0.9512 0 0.9000

16.0000 -0.0000 0.9512 0 0.900017.0000 -0.0000 0.9512 0 0.9000

18.0000 -0.0000 0.9512 0 0.9000

19.0000 -0.0000 0.9512 0 0.9000

20.0000 -0.0000 0.9512 0 0.9000

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Introduction to Numerical AnEngineers

Fundamentals of Digital Compu Digital Computer Models

Convergence, accuracy and stabilit

Number representation

Arithmetic operations Recursion algorithms

Error Analysis Error propagation numerical stab

Error estimation

Error cancellation Condition numbers

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m M

b Be E

Floating Number Representati

Examples

Convention

Decimal

Binary

Decimal

Binary

General

Max mantissa

Min mantissa

Max exponent

Min exponent

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Error Analysis

Number Representation

Absolute Error

Relative Error

Addition

Multiplic

R

A

R

Shift mantis

Result has ex

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Error Propagation

Spherical Bessel Functions

Differential Equation

Solutions

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Error Propagation

Spherical Bessel Functions

Forward Recurrence

Backward Recurrence

Forward Recurrence

Millers algorithm

N ~ x+20

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Error Propagation

Eulers MethodDifferential Equation

Example

Discretization

Finite Difference (forward)

Recurrence

Central Finite Difference

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Error Propagation

Absolute Errors

Function of one variable

General Error Propagation Formula

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Error Propagation

Example

Multiplication

=>

=>

=>

=>

Error Pr

Relative Errors Add for Multiplication

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Error Propagation

Expectation of Errors

Truncation

Error Expectation

Rounding

Addition

Sta

Standard

of e

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Error Propagation

Error Cancellation


Max. error

Stand. errorE

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Error Propagation

Condition Number

x = x(1 + )

xx

y = y(1

yy

y = f(x)

Problem

Error canc

Problem Condition Number

Well-cond

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Error Propagation

Condition NumberProblem Condition Number

4 Significant Digits

Algorithm Condition Number

K is algorit

may be muc

to limited nu

Solution

Higher pre

Rewrite alg

A

Well-con

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Systems of Linear Equations

Mathews Cramers Rule

Gaussian Elimination Numerical implementation

3.3-3.4

Numerical stability

Partial Pivoting Equilibration

Full Pivoting

Multiple right hand sides

Computation count

LU factorization

Error Analysis for Linear Systems

Condition Number

Special Matrices

Iterative Methods Jacobis method

Gauss-Seidel iteration

Convergence

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Systems of Linear EquationCramers Rule

Cramers Rule, n=2

Linear System of Equations

Exa

Cramers rule inco

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Systems of Linear EquationGaussian Elimination

Red

StLinear System of Equations

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Reduction

Step 1

i

j

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Re

StReduction

Step k

Back-S

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Pivotal Elements

Step k Partial Pivo

Row k

Row i

Required at each step!

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Pivotal Elements

Reduction

Step kPartial Pivo

New Row k

New Row i

Required at each step!

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Example, n=2

Cramers Rule - Exact

Gaussian

2-dig

1% error

100% error

n=3

a = [ [0.01 1.0]' [-1.0 0.01]']

b= [1 1]'

r=a^(-1) * b

x=[0 0];m21=a(2,1)/a(1,1);

a(2,1)=0;

a(2,2) = radd(a(2,2),-m21*a(1,2),n);

b(2) = radd(b(2),-m21*b(1),n);

x(2) = b(2)/a(2,2);

x(1) = (radd(b(1), -a(1,2)*x(2),n))/a(1,1);

x'

tbt.m

tbt.m

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Example, n=2


Partial Piv

Interc

2-dig

1% error

1% error

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Example, n=2


2-dig

Multiply E

1% error

100% error

Infinity-NoEquations must be normalized for

partial pivoting to ensure stability

ThisEquilibration is made by

normalizing the matrix to unit norm

Two-Nor

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Example, n=2


Interchan

2-digit A

Pivotin

Full Pivoting

Find largest numerical value in same row and column

Affectsordering of unknowns

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Partial Pivoting

Equilibrate system of equations

Pivoting by Columns

Simple book-keeping

Solution vector in original order

Full Pivoting

Does not require equilibration

Pivoting by both row and columns

More complex book-keeping Solution vector re-ordered


Numerical Stability

Partial Pivoting is simplest and most co

Neither method guarantees stabilit

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Example, n=2


Variable Tra

2-digit A

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System of equations must be well co

Investigate condition number

Tricky, because it requires matrix inversi

Consistent with physics

E.g. dont couple domains that are physi

Consistent units

E.g. dont mix meter and m in unknown

Dimensionless unknowns

Normalize all unknowns consistently

Equilibration and Partial Pivoting, or


How to Ensure Numerical Stab

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Cramers Rule


3.3-3.4

Numerical stability


Full Pivoting


Computation count

LU factorization


Condition Number

Special Matrices



Convergence

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k pn-k

Reduction

Step k

n

k

Compu

Reduc

Total Com

Re

Back

Multiple Right-hand Sides

Reduction for each right-hand side inefficient.

However, RHS may be result of iteration and unknown a priori

(e.g. Eulers method) ->LU Factorization

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Systems of Linear EquationLU Factorization

The coefficient Matrix A is decomposed as

where is a lower triangular matrix

and is an upper triangular matrix

Then the solution is performed in two simple steps

1.

2.

Forward substitution

Back substitution

How to determine and ?

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After reduction step i-1:

i

j

Above and on diagonal

Below diagonal

Unchanged after step i-1

Become and remain 0 in step j


Change in re

Total chan

Total cha

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Matrix product

= xi

j

Below diagonal

k

k

Sum stops at diagonal

00

= x

i

j

Above diagonal

k

k

00

Lower triangular

0

0

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=

=

Upper triangular

Lower triangular

GE Reduction directly yields LU factorization

Comp

Lower diag

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Systems of Linear EquationPivoting in LU Factorizatio

Before reduction, step k

Pivoting if

else

Interchange rowsi andk

Pivot element vector

Forward s

Intercha

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Linear Systems of EquationError Analysis

Lin

How is th

depende

Small changes i

The syste


Condition number

The condition number K is a

measure of the amplification of the

relative errorby the function f(x)

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Vector and Matrix Norm

Properties

Perturbed R

Subtract or

Relative Err

Conditio

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Vector and Matrix Norm

Properties

Perturbed C

Subtract unpe

Relative Err

Conditio

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Ill-Conditioned System

n=4

a = [ [1.0 1.0]'

b= [1 2]'

ai=inv(a);

a_nrm=max( abs(a

abs(a

ai_nrm=max( abs(a

abs(ai

k=a_nrm*ai_nrm

r=ai * b

x=[0 0];

m21=a(2,1)/a(1,1)

a(2,1)=0;

a(2,2) = radd(a(

b(2) = radd(b(2

x(2) = b(2)/a(2

x(1) = (radd(b

x'

Ill-conditioned system

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Well-Conditioned System

n=4

a = [ [0.0001 1.0

b= [1 2]'

ai=inv(a);

a_nrm=max( abs(a

abs(a

ai_nrm=max( abs(a

abs(a

k=a_nrm*ai_nrm

r=ai * b

x=[0 0];

m21=a(2,1)/a(1,1)

a(2,1)=0;

a(2,2) = radd(a(

b(2) = radd(b(2

x(2) = b(2)/a(2

x(1) = (radd(b

x'

Well-conditioned system

4-di

Algorithm

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Cramers Rule


3.3-3.4

Numerical stability


Full Pivoting


Computation count

LU factorization


Condition Number

Special Matrices



Convergence

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Linear Systems of EquationTri-diagonal Systems

y(x,t)x i

Forced Vibration of a Stringf(x,t)

Harmonic excitation

f(x,t) = f(x) cos(t)


Boundary Conditions

Discrete Differen

Matrix F

Tridiagonal Ma

Finite Diffe

Symmetric, positive

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General Tri-diagonal Systems

LU Factorization

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LU FactorizationRedu

Forward

Back Su

LU Factorization:

Forward substitution:Back substitution:

Total:

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Linear Systems of EquationSpecial Matrices

General, Banded Coefficient Matrix

0

0

p

q

p super-diagonals

q sub-diagonals

w = p+q+1 bandwidth

b is half-band

Banded Symme

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0

0

q

0

p


Banded Coefficient Matrix

Gaussian EliminationNo Pivoting

= =

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0

0

q

=


Gaussian EliminationWith Pivoting

q


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0

0

q

q

p


Compact Storage


0

n2

Diagon

n(p

i

j

i

j - i

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Sparse and Banded Coefficient

Skyline Systems

..

Skyline storage applicable when no pivoting is needed, e

symmetric, and positive definite matrices: FEM and FD m

solvers are usually based on Choleski factorization

Skyline

Storage

Pointers 1 4 9 11 16 2

0

0

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Symmetric, Positive Definite Coefficient MaNo pivoting needed

Choleski Factorization

where

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Cramers Rule


3.3-3.4

Numerical stability


Full Pivoting


Computation count

LU factorization


Condition Number

Special Matrices



Convergence

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Linear Systems of EquationIterative Methods

x

x

x

xx

xx

x x

x

0

0

0

0

0

0

Sparse, Full-bandwidth Systems

00

0

0

0

0

0

0

0

Rewrite Equ

Iterative, Recursi

Gauss-Seidels

Jacobis M

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Convergence

Iteration Matrix form

Decompose Coefficient Matrix

with

/

/

Jacobis

Iteration M

Convergen

Sufficient Conve

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Sufficient Convergence Condition

Sufficient Convergence Condition

Jacobis Method

Diagonal Dominance

Stop Criter

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vib_string.m

n=99;

L=1.0;

h=L/(n+1);k=2*pi;

kh=k*h

x=[h:h:L-h]';

a=zeros(n,n);

f=zeros(n,1);

o=1

a(1,1) =kh^2 - 2;

a(1,2)=o;

for i=2:n-1a(i,i)=a(1,1);

a(i,i-1) = o;

a(i,i+1) = o;

end

a(n,n)=a(1,1);

a(n,n-1)=o;

nf=round((n+1)/3);

nw=round((n+1)/6);

nw=min(min(nw,nf-1),n-nf);

figure(1)hold off

nw1=nf-nw;

nw2=nf+nw;

f(nw1:nw2) = h^2*hanning(nw2-nw1+1);

subplot(2,1,1); plot(x,f,'r');

% exact solution

y=inv(a)*f;

subplot(2,1,2); plot(x,y,'b');

% Iterative solution using Jacobi

b=-a;

c=zeros(n,1);

for i=1:n

b(i,i)=0;

for j=1:n

b(i,j)=b(i,j)/a(i,i);

c(i)=f(i)/a(i,i);

end

end

nj=100;

xj=f;

xgs=f;

figure(2)

nc=6

col=['r' 'g' 'b' 'c' 'm' 'y']

hold off

for j=1:nj

xj=b*xj+c;

xgs(1)=b(1,2:n)*xgs(2:n) + c(

for i=2:n-1

xgs(i)=b(i,1:i-1)*xgs(1:i

endxgs(n)= b(n,1:n-1)*xgs(1:n-1)

cc=col(mod(j-1,nc)+1);

subplot(2,1,1); plot(x,xj,cc)

subplot(2,1,2); plot(x,xgs,cc

hold on

end

Off-diagonal values

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vib_string.mo=1.0

Exact Solution It

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vib_string.mo = 0.5

Exact Solution It

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Roots of Non-linear Equations

Herons formula

Stop criteria

General method

Convergence

Examples

Newton-Raphsons Method

Convergence Speed

Examples

Secant Method

Convergence and efficiency

Examples

Multiple roots

Bisection

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Roots of Nonlinear Equatio

Example Square root

Herons Principle

a=2;

n=6;

g=2;

% Number of Digits

dig=5;

sq(1)=g;

for i=2:n

sq(i)= 0.5*radd

end

' i v

[ [1:n]' sq']

hold off

plot([0 n],[sqrt

hold on

plot(sq,'r')

plot(a./sq,'r-.

plot((sq-sqrt(a)

grid on

i value

1.0000 2.0000

2.0000 1.5000

3.0000 1.4167

4.0000 1.4143

5.0000 1.4143

6.0000 1.4143

Guess root

Mean is better guess

Iteration Formula

( )/2

( )/2

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Roots of Nonlinear EquatioStop-criteria

Unrealistic stop-criteria

Realistic stop-criteria

Machine

Accuracy

x

f(x)

flat f(x)

f(x)

steep f(x)

Cannot require

Use combinatio

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Roots of Nonlinear EquatioGeneral Method

Non-linear Equation

Goal: Converging series

Rewrite Problem

Example

Iteration

% f(x) = x^3 - a

% g(x) = x + C*(x

a=2;

n=10;

g=1.0;

C=-0.1;

sq(1)=g;

for i=2:n

sq(i)= sq(i

end

hold off

plot([0 n],[

hold on

plot(sq,'r')

plot( (sq-a^

grid on

Exam

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Convergence

x

y

Define

Apply

Co

Conve

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y

y

Mean-value Theorem

x

x x

1

10

y=g(x)

y

Convergence

Convergence

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Example: Cube root

Rewrite

Convergence

Converges more rapidly for small

n=10;

g=1.0;

C=-0.21;sq(1)=g;

for i=2:n

sq(i)= sq(i-1)

end

hold off

f=plot([0 n],[a^

set(f,'LineWidth

hold on

f=plot(sq,'r')

set(f,'LineWidth

f=plot( (sq-a^(1

set(f,'LineWidth

legend('Exact','

f=title(['a = '

set(f,'FontSize'

grid on

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Converging, but how close?

General Convergence Rule

Absolute error

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Roots of Nonlinear EquatioNewton-Raphson Method

Newton-Raphson Iteration

Fast Convergence

Convergence Criteria

Non-linear Equation

f(x)

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Example Square Root

Newton-Raphson

Same as Herons formula

a=26;

n=10;

g=1;

sq(1)=g;for i=2:n

sq(i)= 0.5*(

end

hold off

plot([0 n],[s

hold on

plot(sq,'r')

plot(a./sq,'r

plot((sq-sqrt

grid on

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a=10;

n=10;g=0.19;

sq(1)=g;

for i=2:n

sq(i)=sq(i-1) -

end

hold off

plot([0 n],[1/a 1

hold on

plot(sq,'r')

plot((sq-1/a)*a,'

grid on

legend('Exact','I

title(['x = 1/' n

Newton-Raphson

Approximate Guess

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Convergence Speed

Taylor Expansion

Second Order Expansion

Relative Error

General Convergence Rate

Conv

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Roots of Nonlinear EquatioSecant Method

1. In Newton-Raphson we have to evaluate 2 functions

2. may not be given in closed, analytical form, i.e. it m

result of a numerical algorithm

Approximate Derivative

Secant Method Iteration

Only 1 function call per iteration:

f(x)

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Roots of Nonlinear EquatioSecant Method

Convergence Speed

Absolute Error

Taylor Series 2nd order

Relative Error

Error E

Error improvement fo

Newton-Raphson

Secant Method

Exponents called

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Roots of Nonlinear EquatioMultiple Roots

f(x)

Newton-Raphson

=>

Convergence

Slower convergence the higherthe order of the root

p-order Root

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Roots of Nonlinear EquatioBisection

f(x)

Algorithm

n = n+1

yes

no Less efficient tha

Secant methods

interval with root

value. Then follow

method for accur

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Roots of Nonlinear EquatioBisection

Algorithm

n = n+1

yes

no

% Root finding by bi

f=inline(' a*x -1','

a=2

figure(1); clf; hold

x=[0 1.5]; eps=1e-3;

err=max(abs(x(1)-x(2

while (err>eps & f(x

xo=x; x=[xo(1) 0

if ( f(x(1),a)*f

x=[0.5*(xo(1

end

x

err=max(abs(x(1)-

b=plot(x,f(x,a),

grid on;

end

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Interpolation 4.1-4

Lagrange interpolation 4.3

Triangular families 4.4

Newtons iteration method 4.4

Equidistant Interpolation 4.4

Numerical Differentiation 6.1-6

Numerical Integration 7.1-7

Error of numerical integration

Mathe

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Numerical Interpolation

Purpose of numerical Interpolatio

1. Compute intermediate values of a samp

2. Numerical differentiation foundation fo

Difference and Finite Element methods

3. Numerical Integration

Given:

Find for

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Numerical InterpolationPolynomial Interpolation

x

f(x)

Interpolation function

Interpolation

Polynomia

Coefficients: Linea

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Linear Interpolation

Examples

x

f(x)f(x)

Quadratic

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Taylor Series

Remainder

Requirement

x

f(x)

f(x)

p(x)

Ill-conditioned for large n

Polynomial is uniq

we calculate the

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Numerical InterpolationLagrange Polynomials

f(x)

1

k-3

Difficult tDifficult t

Divisions

Importan

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Numerical InterpolationTriangular Families of Polynom

Ordered Polynimials

where

Special form c

for interpo

Coefficie

foun

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Numerical InterpolationTriangular Families of Polynom

Polynomial EvaluationHorners Scheme

Remainder Interpolation Error

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Numerical InterpolationNewtons Iteration Formul

Standard triangular family of polynomials

Divided Differences

Newtons

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x

f(x) f(x)

01 2 3 4

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Numerical InterpolationEquidistant Newton Interpola

Divided

Stepsiz

Equidistant Sampling

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function[a] = interp_test(n)

%n=2

h=1/n

xi=[0:h:1]

f=sqrt(1-xi.*xi) .* (1 - 2*xi +5*(xi.*xi));

%f=1-2*xi+5*(xi.*xi)-4*(xi.*xi.*xi);

c=newton_coef(h,f)

m=101

x=[0:1/(m-1):1];

fx=sqrt(1-x.*x) .* (1 - 2*x +5*(x.*x));

%fx=1-2*x+5*(x.*x)-4*(x.*x.*x);

y=newton(x,xi,c);

hold off; b=plot(x,fx,'b'); set(b,'LineWidth',2);hold on; b=plot(xi,f,'.r') ; set(b,'MarkerSize',30);

b=plot(x,y,'g'); set(b,'LineWidth',2);

yl=lagrange(x,xi,f);

b=plot(x,yl,'xm'); set(b,'Markersize',5);

b=legend('Exact','Samples','Newton','Lagrange')

b=title(['n = ' num2str(n)]); set(b,'FontSize',16);

function[y] = newton(x,xi,c)

% Computes Newton polynomial

% with coefficients cn=length(c)-1

m=length(x)

y=c(n+1)*ones(1,m);

for i=n-1:-1:0

cc=c(i+1);

xx=xi(i+1);

y=cc+y.*(x-xx);

end

function[c] = newton_coef(h,f)

% Computes Newton Coefficients

% for equidistant sampling hn=length(f)-1

c=f; c_old=f; fac=1;

for i=1:n

fac=i*h;

for j=i:n

c(j+1)=(c_old(j+1)-c_old(j))/fac;

end

c_old=c;

end

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Interpolation 4.1-4

Lagrange interpolation 4.3

Triangular families 4.4

Newtons iteration method 4.4

Equidistant Interpolation 4.4

Numerical Differentiation 6.1-6

Numerical Integration 7.1-7

Error of numerical integration

Mathe

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Numerical Differentiation

Taylor Series

First order

f(x)

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f(x)

Numerical Differentiation

Second order

Second Derivatives

n=3 Central Difference

Forward Differencen=2

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Numerical Integration

f(x)

a

Lagrange Interpolation

Equidistant Sampling

Integration Weights (Cotes Numbers)

Properties

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Numerical Integration

Trapezoidal Rule

Simpsons Rule

f(x)

f(x)

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Numerical IntegrationError Analysis

x

f(x)

h h

Trapezoidal Rule

Local Absolute Error

Global Error

S

Local ErrorN Intervals

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Ordinary Differential Equations

Initial Value Problems

Eulers Method

Taylor Series Methods

Error analysis

Runge-Kutta Methods

Systems of differential equations

Boundary Value Problems

Shooting method

Direct Finite Difference methods

M

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Ordinary Differential Equation



non-linear in y

Non-Linear Differential Equation

Linear Differential Equation

Linear differential equations can often be solv

Non-linear equations require numerical soluti

y

a

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Eulers Method


Example

Discretization

Finite Difference (forward)

Recurrence

Ordinary Differential Equati


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Taylor Series

Partial Derivatives

Derivatives

+

Truncate ser

Initial Value Problem

Choose

Recursio

w

Loca

Discr

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Example

Eulers Method

General Taylor Series Method Example

Err

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Error Analysis De

Error Estimates and Convergence

Eulers Method

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Error AnalysisExample Eulers Method

Exact solution

Derivative Bounds

Error Bound

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Initial Value ProblemsRunge-Kutta Methods

Taylor Series Recursion

Runge-Kutta Recursion

Matcha,b, to match Taylor series amap.

Substitute k

Match 2nd ord

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Initial Value ProblemsRunge-Kutta Methods

Initial Value Problem

2nd Order Runge-Kutta

4th Order Runge-Kutta

y

mid-poi

Predic

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Eulers Method

Recurrence


Runge-Kutta Methodsh=1.0;

x=[0:0.1*h:10];

y0=0;

y=0.5*x.^2+y0;

figure(1); hold offa=plot(x,y,'b'); set

% Euler's method, fo

xt=[0:h:10]; N=lengt

yt=zeros(N,1); yt(1)

for n=2:N

yt(n)=yt(n-1)+h*

end

hold on; a=plot(xt,y

% Runge Kutta

fxy='x'; f=inline(fx

[xrk,yrk]=ode45(f,xt

a=plot(xrk,yrk,'.g')

a=title(['dy/dx = '

set(a,'FontSize',16)

b=legend('Exact',['E

'Runge-Kutta (Matlab

4th Order Runge-Kutta

Matlab ode45 automatically ensures convergence

Matlab inefficient for large problems > Convergence Analysis

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Ordinary Differential Equations


Eulers Method


Error analysis

Runge-Kutta Methods

Systems of differential equations


Shooting method

Direct Finite Difference methods

M

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Higher Order Differential Equ


Initial

Conditions

Convert to 1s

Matr

Solved using e.g

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Shooting Method


y

a

Boundary

Conditions

Shooting Method

Initial value Problem

Solve by Runge-Kutta

Shooting Iteration

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Direct Finite Difference Metho


y

a

Boundary

Conditions

Discretization

Finite Differences

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Direct Finite Difference MethoBoundary value Problem

Finite Differences

Substitute Finite Differences

Difference Equations

N-1 equations, N-1 unknowns

Matrix

Linear Differen

Solve using standard l

-1

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Boundary Value ProblemsFinite Difference Methods

Forced Vibration of a Stringf(x,t)

Harmonic excitation

f(x,t) = f(x) cos(t)


Boundary Conditions

Discrete Differenc

Matrix F

Tridiagonal Mat

Finite Diffe

Symmetric, positive d

y(x,t)x i

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Boundary Value ProblemsFinite Difference Methods

Boundary Conditions with Derivatives

Difference Equations

Centra

N-1

General Bou

Add extra point - N

Finite DiffereO(h )4

O(h )2

Central Difference

Backward Difference

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Minimization Problems

Least Square Approximation

Normal Equation

Parameter Estimation Curve fitting

Optimization Methods

Simulated Annealing

Traveling salesman problem

Genetic Algorithms

M

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Minimization Problems

Data Modeling Curve Fitting

Mini

N

Objective: Find c that minimizes error

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Least Square Approximatio

m

n

m

m

n

n model parameters

m measurements

Linear Measurement Model Overd

Lea

Minim

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Least Square Approximatio

A

Theorem

)

Proof

Nor

Symmetr

singular

linea

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% Quadratic data model

fxy='a*x.^2+b'

f=inline(fxy,'x','a','b');

x=[0:0.01:1]; x=reshape([x' x']',1,2*length(x));

n=length(x); y=zeros(n,1);

a=5; b=3;

% Generate noisy data

amp=0.05*(max(f(x,a,b))-min(f(x,a,b)));

for i=1:n

y(i) =f(x(i),a,b)+random('norm',0,amp);

end

figure(1); clf; hold off; p=plot(x,y,'.r');

set(p,'MarkerSize',10)

% Non-linear, quadrati model

A=ones(n,2); A(:,1)=f(x,1,0)'; bb=y;

%Normal matrix

C=A'*A; c=A'*bb;

z=inv(C)*c

% Residuals

r=bb-A*z; rn=sqrt(r'*r)/n

hold on; p=plot(x,f(x,z(1),z(2)),'b'); set(p,'LineWidth',2)

% Linear model

A(:,1)=x';

C=A'*A; c=A'*bb;

z=inv(C)*c

% Residuals

r=bb-A*z; rn=sqrt(r'*r)/n

hold on; p=plot(x,z(1)*x+z(2),'g'); set(p,'LineWidth',2)

p=legend('Data','Non-linear','Linear'); set(p,'FontSize',14);

Least Square ApproximatioCurve Fitting

curve.m

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Optimization ProblemsNon-linear Models

Minimim

x

E(c)

Measured

Non

i

Global

Minimum

Local

Minimum

Non-linear models often have multiple, local minima. A locally

approximation may therefore find a local minimum instead of th

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Simulated AnnealingExample: Traveling Salesman Pr

Objective:

Visit N cities acorder, in the sh

Metr

1. Configura

can vary2. Rearrang

any two c

3. Cost func

number o

4. Annealin

with coo

for e.g. 1

method).

Penalty for crossing Mississippi

Cost function: Distance Traveled

East: = 1

West: = -1

Adapted by MIT OCW.

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Simulated AnnealingExample: Traveling Salesman Pr

% Travelling salesman problem

% Create random city distribution

n=20; x=random('unif',-1,1,n,1); y=random('unif',-1,1,n,1);

gam=1; mu=sign(x);% End up where you start. Add starting point to end

x=[x' x(1)]'; y=[y' y(1)]'; mu=[mu' mu(1)]';

figure(1); hold off; g=plot(x,y,'.r'); set(g,'MarkerSize',20);

c0=cost(x,y,mu,gam); k=1; % Boltzman constant

nt=50; nr=200; % nt: temp steps. nr: city switches each T

cp=zeros(nr,nt);

iran=inline('round(random(d,1.5001,n+0.4999))','d','n');

for i=1:nt

T=1.0 -(i-1)/nt

for j=1:nr

% switch two random cities

ic1=iran('unif',n); ic2=iran('unif',n);

xs=x(ic1); ys=y(ic1); ms=mu(ic1);

x(ic1)=x(ic2); y(ic1)=y(ic2); mu(ic1)=mu(ic2);

x(ic2)=xs; y(ic2)=ys; mu(ic2)=ms;

p=random('unif',0,1); c=cost(x,y,mu,gam);

if (c < c0 | p < exp(-(c-c0)/(k*T))) % accept

c0=c;

else % reject and switch back

xs=x(ic1); ys=y(ic1); ms=mu(ic1);

x(ic1)=x(ic2); y(ic1)=y(ic2); mu(ic1)=mu(ic2);

x(ic2)=xs; y(ic2)=ys; mu(ic2)=ms;

end

cp(j,i)=c0;

end

figure(2); plot(reshape(cp,nt*nr,1)); drawnow;

figure(1); hold off; g=plot(x,y,'.r'); set(g,'MarkerSize',20);

hold on; plot(x,y,'b');

g=plot(x(1),y(1),'.g'); set(g,'MarkerSize',30);

p=plot([0 0],[-1 1],'r--'); set(g,'LineWidth',2); drawnow;

end

salesman.m function n=length(

c=0;

for i=1:nc =c+s

+ g

end

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

y

Plot of x*sin(x) vs. x

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

y

y1 = x sin(x)

y2 = sin(x) _._

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

y

y2 = sin(x) _._

y1 = x sin(x)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

y

Plot of sin(x) vs. x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

y

Plot of x*sin(x) vs. x

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

ty(t)=sin(alpha*t)

simple plot by jleonard 2/23/97

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90 60 30 0 30 60 9080

70

60

50

40

30

20

10

0f = 120 kHz D = 0.037 m beamwidth = +9.899 deg

theta (degrees)

Normalizedsourcelevel(dB)

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Issued:

February

3rd,

2005

13.002 IntroductiontoNumericalMethods forEngineers

In-classprogrammingexercises

1. Letabeapositiverealnumber,andletthesequenceofrealnumbersxi begivenby

1 ax0 =1, xi+1 =

2(xi

+xi

),

fori=0, 1, 2, 3, . . ..

Thevaluexi willconvergeto

aasiWriteaprogramthatreadsinthevalueofa

interactivelyandusesthisalgorithmtocomputethesquarerootofa.

Test your program as you vary the maximum number of iterations of the algorithm isincreased from1, 2, 3, . . .anddeterminehowmanysignificantdigitsofprecisionthatyouobtain

for

each.

How

many

iterations

are

necessary

to

reach

the

machine

precision

of

matlab?

2. Writeaprogramtoevaluateebytheseries:

1 1 1 1e=1+1+ + + + +. . .

2! 3! 4! 5!

Testyourprogramasyouincreasethenumberoftermsintheseries. Determinehowmanysignificantdigitsofprecisionthatyouobtain inyouranswerasa functionofthenumberoftermsintheseries. Howmanytermsarenecessarytoreachmachineprecision?

3. Considerthefunctionx sin(x)1.

(a.) Howmanyrootsdoesthisfunctionhaveintheinterval[0, ]?

(b.) Write a matlab program to find the root(s) using Newton-Raphson iteration withappropriatestartingvalues.

(c.)

Make

a

graph

of

relative

error

vs.

iteration

step

for

all

roots.

(d.)

How

many

iterations

are

needed

to

reach

an

error

of

less

than

108?

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13.002J/10.002J

Introduction

to

Numerical

Analysis

for

Engineers

Problem

Set

1

Issued:

February

3,

2005

Due:

February

10,

2005

Do

the

following

problems

from

Mathews

and

Fink:

1.2.4(a)and(b)(onpage23)

1.2.5(a)and(b)

1.2.13(b)(onpage24)

1.3.1(b)and(c)(onpage37)

1.3.12(onpage39)

1.3.13

(a)

Programmingexercise1(onpage39)

Programmingexercise2

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13.002

PS #1 Solution

1.2.4. (a)

(1.0110101)2 = (2^0)+(2^-2)+(2^-3)+(2^-5)+(2^-7)= 1.41406250000000

1.2.4. (b)

(11.0010010001)2 = (2^1)+(2^0)+(2^-3)+(2^-6)+(2^-10)= 3.14160156250000

1.2.5. (a)

2 (1.0110101) = 1.41421356237309 1.41406250000000=2

= 1.510623730900385e-004 (absolute error)

1.510623730900385e-004= 1.068172283940988e-004 (relative error)

1.41421356237309

1.2.5. (b)

(11.0010010001)2 = 3.14159265358979- 3.14160156250000

= -8.908910209992627e-006 (absolute error)

8.908910209992627e-006= 2.835794194964366e-006 (relative error)

3.14159265358979

1.2.13. (b)

1 1 1( + ) + =10 3 5

21((0.1101) 23 + (0.1011) ) = (0.1110) 212 2 2

22(0.1110) 21 + (0.1101) = (0.1010) 202 2 2

19 (0.1010) 20

302

0.63333333333333- 0.6250000000000= 0.0083333333333 (absolute error)

0.0083333333333= 0.01315... (relative error)

0.63333333333333

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1.3.1.(b)

98350-98000 = 350 (absolute error)

350= 0.00355871886121 (relative error) (2 significant digits)

98350

1.3.1.(c)

0.000068-0.00006= 8e-006 (absolute error)

8e-006= 0.117647058 (relative error) (no significant digits)

0.000068

1.3.12

b2

b2

b + 4ac b + 4ac 2c =x1new =

b2

b22a b + 4ac b + 4ac

b2

b2

b 4ac b 4ac 2c =x2new =

b2

b22a b 4ac b 4ac

1.3.13. (a)

2x 1, 000.001x +1 = 0

b2

b + 4acx1 = = 1000

2a

2cx2 = = 0.001

b2

b 4ac

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Programming Exercise 1

% script M-file findroots.m

a=input(Enter the value of "a" from ax^2+bx+c=0 :);

b=input(Enter the value of "b" from ax^2+bx+c=0 :);

c=input(Enter the value of "c" from ax^2+bx+c=0 :);if b>= 0;

sign=1;

else sign=-1;

end;

q=-0.5*(b+sign*sqrt((b^2)-4*a*c));

x1=q/a;

x2=c/q;

xx1=num2str(x1);

xx2=num2str(x2);

disp([X1 is equal to , xx1])

disp([X2 is equal to ,xx2])

Programming Exercise 2

x(1)=1/2;r(1)=0.994; p(1)=1; p(2)=0.497; q(1)=1; q(2)=0.497;

for n=2:11

x(n)=(1/2)*x(n-1);

end

for n=2:11

r(n)=(1/2)*(r(n-1));

end

for n=3:11

p(n)=(3/2)*p(n-1)-(1/2)*p(n-2);

end

for n=3:11q(n)=(5/2)*q(n-1)-q(n-2);

end

h=1:11;

figure(1)

plot(h, x(h)-r(h),bd,h, x(h)-p(h),r+,h, x(h)-q(h),g)

grid on

legend(r(n),p(n),q(n))

fprintf(n x(n) r(n) p(n) q(n)\n)

for i = h

fprintf(%2d %+10.8f %+10.8f %+10.8f %+10.8f\n, i, x(i), p(i), r(i), q(i))

endfprintf( n x(n)-r(n) x(n)-p(n) x(n)-q(n)\n)

for i = h

fprintf(%2d %+10.8f %+10.8f %+10.8f\n, i, x(i)-r(i), x(i)-p(i), x(i)-q(i))

end

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n x(n)

1 +0.50000000

2 +0.25000000

3 +0.12500000

4 +0.06250000

5 +0.03125000

6 +0.01562500

7 +0.00781250

8 +0.00390625

9 +0.00195313

10 +0.00097656

11 +0.00048828

n x(n)-r(n)

1 -0.49400000

2 -0.24700000

3 -0.12350000

4 -0.06175000

5 -0.03087500

6 -0.015437507 -0.00771875

8 -0.00385938

9 -0.00192969

10 -0.00096484

11 -0.00048242

>>

r(n) p(n)

+1.00000000 +0.99400000

+0.49700000 +0.49700000

+0.24550000 +0.24850000

+0.11975000 +0.12425000

+0.05687500 +0.06212500

+0.02543750 +0.03106250

+0.00971875 +0.01553125

+0.00185938 +0.00776563

-0.00207031 +0.00388281

-0.00403516 +0.00194141

-0.00501758 +0.00097070

q(n)

+1.00000000

+0.49700000

+0.24250000

+0.10925000

+0.03062500

-0.03268750

-0.11234375

-0.24817188

-0.50808594

-1.02204297

-2.04702148

x(n)-p(n)

-0.50000000

-0.24700000

-0.12050000

-0.05725000

-0.02562500

-0.00981250-0.00190625

+0.00204687

+0.00402344

+0.00501172

+0.00550586

x(n)-q(n)

-0.50000000

-0.24700000

-0.11750000

-0.04675000

+0.00062500

+0.04831250+0.12015625

+0.25207813

+0.51003906

+1.02301953

+2.04750977

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Amendment to Problem Set 4

The previous equation has been changed to

be:

x1

2

3

1 0

2 e

1 1 0

1 1

2 1

x1

x2 = 2

x3

= 6

01.

1

e

e

e

=

xx

1

1

1

=

x1x

2

x3

1e1

1

=

1

1

3=

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=

x3

=

2

2.

clear

clc

alfa=100;

A=[exp(-alfa) 1 0; -1 exp(-alfa) -1; 1 -2

exp(-alfa)];

%A=[1 exp(-alfa) 0; exp(-alfa) -1 -1; -2 1

exp(-alfa)];oA=A;

B=[1; exp(-alfa); exp(-alfa)];

oB=B;

x2

x1

x3

0 1 0

1 0

2 0

1

0

=

0

1

1

2

1

=

=

x1

x2

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[mm,n]=size(A);

A=[A B];L=zeros(mm,n);

for i=1:mm-1

for j=i+1:mm

m(j,i)=A(j,i)/A(i,i);

L(j,i)=m(j,i);for k=i:n+1

A(j,k)=A(j,k)-m(j,i)*A(i,k);

end

end

end

U=A(:,1:n);

L=L+eye(mm,n);

B=A(:,n+1);

x=zeros(n,1);

for j=mm:-1:1x(j)=(B(j)-A(j,j+1:n)*x(j+1:n))/A(j,j);

end

x

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oB-oA*x

3.

=0

1

=2

2

= 63

x

x

x

=3

AxB =

0 The solutions are exactly correct.

=5

1

= 9866.02

x

x

x3

= 9934.1

=

9933.1

AxB 0 Solutions are very good

approximation.

=10

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1 =2

= 9999.02

=

2

3

x

x

x

AxB 0 Solutions are very goodapproximation

=

20

=21

=12

= 23

x

x

x

AxB

0 Solutions are very goodapproximation

=

40

=0x

x

x

1

=12

=03

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0

AxB

=

0

2

Big errors exist. Without

pivoting, its not stable now.

4.

clearclc

alfa=40;

%A=[1 exp(-alfa) 0; exp(-alfa) -1 -1; -2 1

exp(-alfa)];A=[exp(-alfa) 1 0; -1 exp(-alfa) -1; 1 -2

exp(-alfa)]; %original one.

oA=A;

B=[1; exp(-alfa); exp(-alfa)];

oB=B;

[mm,n]=size(A);

A=[A B];

L=zeros(mm,n);

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for i=1:mm-1

%pivoting begins from here.

[Y I]=max(abs(A(i:mm,i)));

temp_store1=A(I,:);

temp_store2=B(I);

A(I,:)=A(i,:);

B(I)=B(i);A(i,:)=temp_store1;

B(i)=temp_store2;

%pivoting ends from here.

for j=i+1:mm

m(j,i)=A(j,i)/A(i,i);

L(j,i)=m(j,i);

for k=i:n+1

A(j,k)=A(j,k)-m(j,i)*A(i,k);

end

endend

U=A(:,1:n);

L=L+eye(mm,n);

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x

B=A(:,n+1);

x=zeros(n,1);for j=mm:-1:1

x(j)=(B(j)-A(j,j+1:n)*x(j+1:n))/A(j,j);

end

oB-oA*x

5. switch x1 and x2

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13.002

Introduction to Numerical Methods for Engineers

Take Home Exam

Issued: Thursday, Mar. 10, 2005

Due: Friday, Mar. 18, 2005

Problem 1.

Problem 2.

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13.002J - Introduction to Numerical Analysis for EngineersClass Survey - Spring 2005

Name:Course:E-mail:WhatprogrammingsubjectshaveyoutakenatMIT?

1.00

6.001

10.001

Self-Assessment Quiz (no grade)

2993 spring 2005

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