ECIV 301 Programming & Graphics Numerical Methods for Engineers REVIEW II.

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ECIV 301 Programming & Graphics Numerical Methods for Engineers REVIEW II

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

Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]

Matrix Algebra Row 1 Row 3 Column 2Column m n x m Matrix

Matrix Algebra 3 rd Row 2 nd Column

Matrix Algebra 1 Row, m Columns Row Vector

Matrix Algebra n Rows, 1 Column Column Vector

Matrix Algebra If n = m Square Matrix e.g. n=m=5 Main Diagonal

Matrix Algebra Special Types of Square Matrices Symmetric: a ij = a ji

Matrix Algebra Diagonal: a ij = 0, i  j Special Types of Square Matrices

Matrix Algebra Identity: a ii =1.0 a ij = 0, i  j Special Types of Square Matrices

Matrix Algebra Upper Triangular Special Types of Square Matrices

Matrix Algebra Lower Triangular Special Types of Square Matrices

Matrix Algebra Banded Special Types of Square Matrices

Matrix Operating Rules - Equality [A] mxn =[B] pxq n=pm=qa ij =b ij

Matrix Operating Rules - Addition [C] mxn = [A] mxn +[B] pxq n=p m=q c ij = a ij +b ij

Matrix Operating Rules - Addition Properties [A]+[B] = [B]+[A] [A]+([B]+[C]) = ([A]+[B])+[C]

Multiplication by Scalar

Matrix Multiplication [A] n x m. [B] p x q = [C] n x q m=p

Matrix Multiplication

Matrix Multiplication - Properties Associative: [A]([B][C]) = ([A][B])[C] If dimensions suitable Distributive: [A]([B]+[C]) = [A][B]+[A] [C] Attention: [A][B]  [B][A]

Operations - Transpose

Operations - Inverse [A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular

Operations - Trace Square Matrix tr[A] =  a ii

Linear Equations in Matrix Form

Gauss Elimination Consider (Eq 1) (Eq 2) Solution 2*(Eq 1) (Eq 2) Solution !!!!!! Scaling Does Not Change the Solution

Gauss Elimination Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) Solution Operations Do Not Change the Solution

Gauss Elimination Example Forward Elimination

Gauss Elimination -

Substitute 2 nd eq with new

Gauss Elimination -

Substitute 3rd eq with new

Gauss Elimination -

Substitute 3rd eq with new

Gauss Elimination Forward Elimination

Gauss Elimination Back Substitution

Gauss Elimination – Potential Problem Pivoting

Partial Pivoting a 32 >a 22 a l2 >a 22 NO YES

Partial Pivoting

Full Pivoting In addition to row swaping Search columns for max elements Swap Columns Change the order of x i Most cases not necessary

LU Decomposition

PIVOTS Column 1 PIVOTS Column 2

LU Decomposition As many as, and in the location of, zeros Upper Triangular Matrix U

LU Decomposition PIVOTS Column 1 PIVOTS Column 2 Lower Triangular Matrix L

LU Decomposition = This is the original matrix!!!!!!!!!!

LU Decomposition Lyb

Lyb

Ax=b A=LU -LU Decomposition Ly=b- Solve for y Ux=y- Solve for x

Matrix Inversion

[A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular

Matrix Inversion

Solution

Matrix Inversion To calculate the invert of a nxn matrix solve n times :

Iterative Methods Recall Techniques for Root finding of Single Equations Initial Guess New Estimate Error Calculation Repeat until Convergence

Gauss Seidel

First Iteration: Better Estimate

Gauss Seidel Second Iteration: Better Estimate

Gauss Seidel Iteration Error: Convergence Criterion:

Jacobi Iteration

First Iteration: Better Estimate

Jacobi Iteration Second Iteration: Better Estimate

Jacobi Iteration Iteration Error:

Determinants Are composed of same elements Completely Different Mathematical Concept

Determinants Defined in a recursive form 2x2 matrix

Determinants Defined in a recursive form 3x3 matrix

Determinants Minor a 11

Determinants Minor a 12

Determinants Minor a 13

Singular Matrices If det[A]=0 solution does NOT exist

Determinants and LU Decomposition

Curve Fitting Often we are faced with the problem… what value of y corresponds to x=0.935?

Curve Fitting Question 1 : Is it possible to find a simple and convenient formula that reproduces the points exactly? e.g. Straight Line ? …or smooth line ? …or some other representation? Interpolation

Curve Fitting Question 2 : Is it possible to find a simple and convenient formula that represents data approximately ? e.g. Best Fit ? Approximation

Linear Interpolation Slope of Line 1 st DIVIDED DIFFERENCE f [x i+1,x i ] First order interpolating polynomial

Function Interpolation Quadratic Interpolation Better Accuracy if 2 nd Order Polynomial x

General Form of Newton’s Interpolating Polynomials

Lagrange Interpolating Polynomials Reformulation of Newton’s Polynomials Avoid Calculation of Divided Differences xf(x) xoxo f(x o ) x1x1 f(x 1 ) x2x2 f(x 2 ) …… xnxn f(x n )

Lagrange Interpolating Polynomial Cardinal Functions: Product of n-1 linear factors Skip x i Property:

Errors in Polynomial Interpolation It is expected that as number of nodes increases, error decreases, HOWEVER…. At all interpolation nodes x i Error=0 At all intermediate points Error: f(x)-f n-1 (x) f(x)

Errors in Polynomial Interpolation Beware of Oscillations…. For Example: Consider f(x)=(1+x 2 ) -1 evaluated at 9 points in [-5,5] And corresponding p 8 (x) Lagrange Interpolating Polynomial P 8 (x) f(x)

Other Methods Direct Evaluation n+1 coefficients n+1 Data Points Interpolating Polynomial should represent them exactly

Other Methods Direct Evaluation

Other Methods Solve Using any of the methods we have learned

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

Inverse Interpolation X=?

X=? Switch x and y and then interpolate? Not a Good Idea!

Splines

Piecewise smooth polynomials

E.G Quadratic Splines Function Values at adjacent polynomials are equal at interior nodes

E.G Quadratic Splines First and Last Functions pass through end points

E.G Quadratic Splines First Derivatives at Interior nodes are equal

E.G Quadratic Splines Assume Second Derivative @ First Point=0

E.G Quadratic Splines Assume Second Derivative @ First Point=0 Solve 3nx3n system of Equations

Spline Interpolation Polynomial Interpolation Spline Interpolation Polynomial Interpolation

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