ECIV 301

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ECIV 301

• Programming Graphics
• Numerical Methods for Engineers
• REVIEW II

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Topics

• Introduction to Matrix Algebra
• Gauss Elimination
• LU Decomposition
• Matrix Inversion
• Iterative Methods
• Function Interpolation Approximation
• Newton Polynomials
• Lagrange Polynomials

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Matrix Algebra
Rectangular Array of Elements Represented by a
single symbol A
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Matrix Algebra
n x m Matrix
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Matrix Algebra
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Matrix Algebra
1 Row, m Columns
Row Vector
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Matrix Algebra
n Rows, 1 Column
Column Vector
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Matrix Algebra
If n m Square Matrix
e.g. nm5
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Matrix Algebra
Special Types of Square Matrices
Symmetric aij aji
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Matrix Algebra
Special Types of Square Matrices
Diagonal aij 0, i?j
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Matrix Algebra
Special Types of Square Matrices
Identity aii1.0 aij 0, i?j
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Matrix Algebra
Special Types of Square Matrices
Upper Triangular
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Matrix Algebra
Special Types of Square Matrices
Lower Triangular
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Matrix Algebra
Special Types of Square Matrices
Banded
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Matrix Operating Rules - Equality
AmxnBpxq
np
mq
aijbij
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Matrix Operating Rules - Addition
Cmxn AmxnBpxq
np
cij aijbij
mq
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Matrix Operating Rules - Addition
Properties
AB BA
A(BC) (AB)C
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Multiplication by Scalar
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Matrix Multiplication
A n x m . B p x q C n x q
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Matrix Multiplication
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Matrix Multiplication
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Matrix Multiplication - Properties
If dimensions suitable
Associative A(BC) (AB)C
Distributive A(BC) ABA C
Attention AB ? BA
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Operations - Transpose
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Operations - Inverse
A
A-1
A A-1I
If A-1 does not exist A is singular
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Operations - Trace
Square Matrix
trA Saii
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Linear Equations in Matrix Form
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Gauss Elimination
Consider
(Eq 1)
2(Eq 1)
(Eq 2)
(Eq 2)
Solution
Solution
!!!!!!
Scaling Does Not Change the Solution
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Gauss Elimination
Consider
(Eq 1)
(Eq 1)
(Eq 2)-(Eq 1)
(Eq 2)
Solution
Solution
!!!!!!
Operations Do Not Change the Solution
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Gauss Elimination
Example
Forward Elimination
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Gauss Elimination
-
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Gauss Elimination
Substitute 2nd eq with new
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Gauss Elimination
-
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Gauss Elimination
Substitute 3rd eq with new
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Gauss Elimination
-
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Gauss Elimination
Substitute 3rd eq with new
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Gauss Elimination
Forward Elimination
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Gauss Elimination
Back Substitution
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Gauss Elimination Potential Problem
Pivoting
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Partial Pivoting
NO
YES
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Partial Pivoting
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Full Pivoting

• In addition to row swaping
• Search columns for max elements
• Swap Columns
• Change the order of xi
• Most cases not necessary

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LU Decomposition
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LU Decomposition
PIVOTS Column 1
PIVOTS Column 2
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LU Decomposition
Upper Triangular Matrix
U
As many as, and in the location of, zeros
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LU Decomposition
PIVOTS Column 2
PIVOTS Column 1
Lower Triangular Matrix
L
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LU Decomposition

This is the original matrix!!!!!!!!!!
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LU Decomposition
L
y
b
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LU Decomposition
L
y
b
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LU Decomposition
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LU Decomposition

• Axb
• ALU - LU Decomposition
• Lyb - Solve for y
• Uxy - Solve for x

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Matrix Inversion
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Matrix Inversion
A
A-1
A A-1I
If A-1 does not exist A is singular
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Matrix Inversion
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Matrix Inversion
Solution
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Matrix Inversion

• To calculate the invert of a nxn matrix solve n
  times

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Iterative Methods
Recall Techniques for Root finding of Single
Equations
Initial Guess New Estimate Error
Calculation Repeat until Convergence
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Gauss Seidel
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Gauss Seidel
First Iteration
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Gauss Seidel
Second Iteration
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Gauss Seidel
Iteration Error
Convergence Criterion
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Jacobi Iteration
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Jacobi Iteration
First Iteration
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Jacobi Iteration
Second Iteration
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Jacobi Iteration
Iteration Error
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Determinants
Are composed of same elements
Completely Different Mathematical Concept
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Determinants
Defined in a recursive form
2x2 matrix
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Determinants
Defined in a recursive form
3x3 matrix
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Determinants
Minor a11
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Determinants
Minor a12
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Determinants
Minor a13
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Singular Matrices
If detA0 solution does NOT exist
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Determinants and LU Decomposition
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Curve Fitting
Often we are faced with the problem
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Curve Fitting
Question 1 Is it possible to find a simple and
convenient formula that reproduces the points
exactly?
Interpolation
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Curve Fitting
Question 2 Is it possible to find a simple and
convenient formula that represents data
approximately ?
Approximation
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Linear Interpolation
First order interpolating polynomial
Slope of Line
1st DIVIDED DIFFERENCE f xi1,xi
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Function Interpolation

• Quadratic Interpolation
• Better Accuracy if
• 2nd Order Polynomial

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General Form of Newtons Interpolating Polynomials
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Lagrange Interpolating Polynomials

• Reformulation of Newtons Polynomials
• Avoid Calculation of Divided Differences

x f(x)
xo f(xo )
x1 f(x1 )
x2 f(x2 )

xn f(xn)
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Lagrange Interpolating Polynomial
Cardinal Functions Product of n-1 linear factors
Property
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Errors in Polynomial Interpolation
f(x)
It is expected that as number of nodes increases,
error decreases, HOWEVER.
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Errors in Polynomial Interpolation
Beware of Oscillations.
For Example Consider f(x)(1x2)-1 evaluated at
9 points in -5,5 And corresponding p8(x)
Lagrange Interpolating Polynomial
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Other Methods
Direct Evaluation
n1 coefficients
n1 Data Points
Interpolating Polynomial should represent them
exactly
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Other Methods
Direct Evaluation
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Other Methods
Solve Using any of the methods we have learned
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Other Methods

• Not the most efficient method
• Ill-conditioned matrix (nearly singular)
• If n is large highly inaccurate coefficients
• Limit to lower order polynomials

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Inverse Interpolation
X?
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Inverse Interpolation
Switch x and y and then interpolate?
X?
Not a Good Idea!
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Splines
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Splines
Piecewise smooth polynomials
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E.G Quadratic Splines

• Function Values at adjacent polynomials are equal
  at interior nodes

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E.G Quadratic Splines

• First and Last Functions pass through end points

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E.G Quadratic Splines

• First Derivatives at Interior nodes are equal

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E.G Quadratic Splines

• Assume Second Derivative _at_ First Point0

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E.G Quadratic Splines

• Assume Second Derivative _at_ First Point0

Solve 3nx3n system of Equations
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Spline Interpolation