Interpolation and Curve Fitting

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Sensitivity <strong>and</strong> Conditioning • Similarly, for perturbation E in matrix A, x x 2 2 cond ( A)(tan( cond ) ( A) 1 ) E A 2 2 • Condition number of least squares solution is about cond(A) if residual is small, but can be squared or arbitrarily worse for large residual Numerical Methods © Wen-Chieh Lin 48
Normal Equation Method • If m × n matrix A has rank n, then symmetric n × n matrix is positive definite, so its Cholesky factorization T T A A LL can be used to obtain solution a to system of normal quations T T A Aa A b which has same solution as linear least squares problem Aa≈b • Normal equations method involves transformations: rectangular square triangular Numerical Methods © Wen-Chieh Lin 49
Page 1 and 2: Interpolation and Curve Fitting Pie
Page 3 and 4: Global vs. Local Control • Does a
Page 5 and 6: Interpolation in Parametric Form
Page 7 and 8: Cubic 3D Curves • Three cubic pol
Page 9 and 10: Hermite Curve Formation— consider
Page 11 and 12: Hermite Curve Formation (cont.) •
Page 13 and 14: Hermite Curve Formation (cont.) •
Page 15 and 16: Blending Functions of Hermite Splin
Page 17 and 18: Bezier Curves (cont.) • Variant o
Page 19 and 20: Composing Bezier Curves • Control
Page 21 and 22: Cubic B-Splines • Basis-spline (B
Page 23 and 24: • From Solving for B-spline Coeff
Page 25 and 26: Blending Functions of Cubic B-Splin
Page 27 and 28: Composing B-splines • Given point
Page 29 and 30: Interpolating on a Surface • Bezi
Page 31 and 32: Numerical Methods © Wen-Chieh Lin
Page 33 and 34: Bezier Surfaces p ( 0,0) p 0,0) 3(
Page 35 and 36: Least Square Approximation • High
Page 37 and 38: Curve Fitting • Given m data poin
Page 39 and 40: Example: curve fitting • Fitting
Page 41 and 42: Example (cont.) • The computed ap
Page 43 and 44: Orthogonality • Space spanned by
Page 45 and 46: Pseudoinverse and Condition Number
Page 47: Sensitivity and Conditioning • Se
Page 51 and 52: Example: Normal Equation Method •

Interpolation and Curve Fitting

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