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

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Matrix Algebra 3 rd Row 2 nd Column

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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. n=m=5 Main Diagonal

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

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

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

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Matrix Algebra Upper Triangular Special Types of Square Matrices

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Matrix Algebra Lower Triangular Special Types of Square Matrices

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Matrix Algebra Banded Special Types of Square Matrices

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

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

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

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Matrix Multiplication

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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]

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Operations - Transpose

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

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

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Linear Equations in Matrix Form

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

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Gauss Elimination Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) 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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Substitute 2 nd eq with new

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Gauss Elimination -

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Substitute 3rd eq with new

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Gauss Elimination -

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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 a 32 >a 22 a l2 >a 22 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 x i Most cases not necessary

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LU Decomposition

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PIVOTS Column 1 PIVOTS Column 2

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

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

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

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LU Decomposition Lyb

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Lyb

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

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Matrix Inversion

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

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Matrix Inversion

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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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First Iteration: Better Estimate

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Gauss Seidel Second Iteration: Better Estimate

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Gauss Seidel Iteration Error: Convergence Criterion:

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Jacobi Iteration

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First Iteration: Better Estimate

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Jacobi Iteration Second Iteration: Better Estimate

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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 a 11

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Determinants Minor a 12

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Determinants Minor a 13

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Singular Matrices If det[A]=0 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… what value of y corresponds to x=0.935?

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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

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

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

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

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General Form of Newton’s Interpolating Polynomials

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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 )

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

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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)

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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)

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Other Methods Direct Evaluation n+1 coefficients n+1 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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X=? Switch x and y and then interpolate? Not a Good Idea!

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Splines

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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 @ First Point=0

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

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Spline Interpolation Polynomial Interpolation Spline Interpolation Polynomial Interpolation

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