Numerical Methods Contents - SAM

More documents

Recommendations

Info

2 1.5 1/(1+x 2 ) Interpolating polynomial 1.2 1 Function f Chebychev interpolation polynomial ||f − p|| 2 2 ≈ b − a 2N ∑ 0≤l
8.5.3 Chebychev interpolation: computational aspects 1.2 1 Function f Chebychev interpolation polynomial ||f−p n || ∞ ||f−p n || 2 ✬ ✩ 0.8 Theorem 8.5.3 (Orthogonality of Chebychev polynomials). 0.6 0.4 0.2 Error norm 10 −1 The Chebychev polynomials are orthogonal with respect to the scalar product ∫ 1 1 〈f,g〉 = f(x)g(x) √ dx . (8.5.7) −1 1 − x 2 ✫ ✪ 0 ✬ ✩ −0.2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 t From the double logarithmic plot, notice • no exponential convergence • algebraic convergence (?) ➙ Error norm 10 0 Polynomial degree n 10 −2 2 4 6 8 10 12 14 16 18 20 ||f−p n || ∞ ||f−p n || 2 10 −1 10 −2 10 0 10 1 10 2 10 0 Polynomial degree n Ôº½ º Theorem 8.5.4 (Discrete orthogonality of Chebychev polynomials). The Chebychev polynomials T 0 , ...,T n are orthogonal in the space P n with respect to the scalar product: n∑ (f, g) = f(x k )g(x k ) , (8.5.8) k=0 where x 0 , ...,x n are the zeros of T n+1 . ✫ ✪ Ôº¿ º ➀ Computation of the coefficients of the interpolation polynomial in Chebychev form: ✬ ✩ ➂ f(t) = { 12 (1 + cos πt) |t| < 1 0 1 ≤ |t| ≤ 2 I = [−2, 2], n = 10 (plot on the left). Theorem 8.5.5 (Representation formula). The interpolation polynomial p of f in the Chebychev nodes x 0 , ...,x n (the zeros of T n+1 ) is given by: p(x) = 1 2 c 0 + c 1 T 1 (x) + ... + c n T n (x) , (8.5.9) 1.2 1 0.8 Function f Chebychev interpolation polynomial 10 −1 ||f−p n || ∞ ||f−p n || 2 with ✫ c k = 2 n + 1 n∑ l=0 ( ( 2l + 1 f cos n + 1 · π )) ( · cos k 2l + 1 2 n + 1 · π ) . (8.5.10) 2 ✪ 0.6 0.4 0.2 Error norm 10 −2 For sufficiently large n (n ≥ 15) it is convenient to compute the c k with the FFT; the direct computation of the coefficients needs (n + 1) 2 multiplications, while FFT needs only O(n log n). 0 −0.2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 t Notice: only algebraic convergence. 10 0 Polynomial degree n 10 −3 10 0 10 1 10 2 ➁ Evaluation of polynomials in Chebychev form: Ôº¾ º ✸ Ôº º
Page 1 and 2:
2 Direct Methods for Linear Systems
Page 3 and 4:
III Integration of Ordinary Differe
Page 5 and 6:
Extra questions for course evaluati
Page 7 and 8:
1.1.2 Matrices Matrices = two-dimen
Page 9 and 10:
Remark 1.2.1 (Row-wise & column-wis
Page 11 and 12:
1.3 Complexity/computational effort
Page 13 and 14:
Syntax of BLAS calls: The functions
Page 15 and 16:
4 { 5 a ssert ( this−>n==B. n &&
Page 17 and 18:
34 long r t 0 ; 35 bool bStarted ;
Page 19 and 20:
Obviously, left multiplication with
Page 21 and 22:
❶: elimination step, ❷: backsub
Page 23 and 24:
A direct way to LU-decomposition:
Page 25 and 26:
Solution of LŨx = b: x ( ) 2ǫ = 1
Page 27 and 28:
numerically equivalent ˆ= same res
Page 29 and 30:
Code 2.4.8: Finding outeps in MATLA
Page 31 and 32:
Terminology: Def. 2.5.5 introduces
Page 33 and 34:
Example 2.5.5 (Instability of multi
Page 35 and 36:
Note: sensitivity gauge depends on
Page 37 and 38:
6 for i =1:20 7 n = 2^ i ; m = n /
Page 39 and 40:
0 20 40 60 80 100 120 140 160 180 2
Page 41 and 42:
Use sparse matrix format: 10 1 10 2
Page 43 and 44:
Envelope-aware LU-factorization: 0
Page 45 and 46:
0 20 40 60 80 100 0 20 0 20 0 20 De
Page 47 and 48:
Evident: symmetry of Ã − bbT a 1
Page 49 and 50:
9 ylabel ( ’ { \ b f c o n d i t
Page 51 and 52:
Mapping a ∈ K n to a multiple of
Page 53 and 54:
Then store G ij (a,b) as triple (i,
Page 55 and 56:
Recall: e i ˆ= i-th unit vector Ch
Page 57 and 58:
Computation of Choleskyfactorizatio
Page 59 and 60:
1 ∃ (partial) cyclic row permutat
Page 61 and 62:
Definition 3.1.3 (Local and global
Page 63 and 64:
Example 3.1.6 (quadratic convergenc
Page 65 and 66:
k |x (k) − π| L 1−L |x (k) −
Page 67 and 68:
(x (k) ) k∈N0 Cauchy sequence ➤
Page 69 and 70:
Termination criterion for contracti
Page 71 and 72:
Given x (k) ∈ I, next iterate :=
Page 73 and 74:
secant method ( MATLAB implementati
Page 75 and 76:
Assuming p = 1: p > 1: ∥ C ∥e (
Page 77 and 78:
This is a simple computation: DG(x)
Page 79 and 80:
k x (k) ǫ k := ‖x ∗ − x (k)
Page 81 and 82:
Code 3.4.14: Damped Newton method (
Page 83 and 84:
MATLAB-CODE: Broyden method (3.4.11
Page 85 and 86:
Algorithm 4.1.3 (Steepest descent).
Page 87 and 88:
Example 4.1.8 (Convergence of gradi
Page 89 and 90:
4.2.1 Krylov spaces Definition 4.2.
Page 91 and 92:
Remark 4.2.3 (A posteriori terminat
Page 93 and 94:
10 figure ; view ([ −45 ,28]) ; m
Page 95 and 96:
Idea: Solve Ax = b approximately in
Page 97 and 98:
eplaced with κ(A) ! 4.4.2 Iteratio
Page 99 and 100:
For circuit of Fig. 55 at angular f
Page 101 and 102:
(Linear) generalized eigenvalue pro
Page 103 and 104:
10 0 10 1 matrix size n d = eig(A)
Page 105 and 106:
0 50 100 150 200 250 300 350 400 45
Page 107 and 108:
1 2 3 k ρ (k) EV ρ (k) EW ρ (k)
Page 109 and 110:
✬ ✩ ✬ ✩ Lemma 5.3.4 (Ncut a
Page 111 and 112:
In other words, roundoff errors may
Page 113 and 114:
Theory: linear convergence of (5.3.
Page 115 and 116:
error in eigenvalue 10 0 10 −2 10
Page 117 and 118:
✬ Residuals r 0 ,...,r m−1 gene
Page 119 and 120: Algebraic view of the Arnoldi proce
Page 121 and 122: 5.5 Singular Value Decomposition Re
Page 123 and 124: Illustration: columns = ONB of Im(A
Page 125 and 126: ✬ Theorem 5.5.7 (best low rank ap
Page 127 and 128: Reassuring: Remark 6.0.4 (Pseudoinv
Page 129 and 130: Consider the linear least squares p
Page 131 and 132: Goal: Euclidean distance of y ∈ R
Page 133 and 134: 6.5 Non-linear Least Squares If (6.
Page 135 and 136: 0 2 4 6 8 10 12 14 16 value of ∥
Page 137 and 138: Definition 7.1.1 (Discrete convolut
Page 139 and 140: Expand a 0 ,...,a n−1 and b 0 , .
Page 141 and 142: (7.2.2) is a simple consequence of
Page 143 and 144: Dominant coefficients of a signal a
Page 145 and 146: 11 c = f f t ( y ) ; 12 13 figure (
Page 147 and 148: Two-dimensional trigonometric basis
Page 149 and 150: 8 end 9 t1 = min ( t1 , toc ) ; 10
Page 151 and 152: Step II: for k =: rq + s, 0 ≤ r <
Page 153 and 154: MATLAB-CODE Sine transform function
Page 155 and 156: △ Example 7.5.2 (Linear regressio
Page 157 and 158: [23, Ch. IX] presents the topic fro
Page 159 and 160: Code 8.1.3: Horner scheme, polynomi
Page 161 and 162: 1.2 equality in (8.2.10) for y := (
Page 163 and 164: ecursive definition: p i (t) ≡ y
Page 165 and 166: a 1 = y 1 − a 0 t 1 − t 0 = y 1
Page 167 and 168: Observations: Strong oscillations o
Page 169: −1 −0.8 −0.6 −0.4 −0.2 0
Page 173 and 174: 9.1 Shape preserving interpolation
Page 175 and 176: 9.2.2 Piecewise polynomial interpol
Page 177 and 178: Interpolation of the function: f(x)
Page 179 and 180: 2 % Plot convergence of approximati
Page 181 and 182: 9.4 Splines Definition 9.4.1 (Splin
Page 183 and 184: ➤ Linear system of equations with
Page 185 and 186: y i+1 t i−1 t i t i+1 y i−1 y i
Page 187 and 188: 35 h= d i f f ( t ) ; 36 d e l t a
Page 189 and 190: 1 0.9 Function f 1 0.9 Function f
Page 191 and 192: f 10 Numerical Quadrature Numerical
Page 193 and 194: f • n = 1: Trapezoidal rule • n
Page 195 and 196: For fixed local n-point quadrature
Page 197 and 198: Equidistant trapezoidal rule (order
Page 199 and 200: Heuristics: A quadrature formula ha
Page 201 and 202: 20 Zeros of Legendre polynomials in
Page 203 and 204: |quadrature error| 10 0 Numerical q
Page 205 and 206: f f • line 9: estimate for global
Page 207 and 208: Model: autonomous Lotka-Volterra OD
Page 209 and 210: Example 11.1.6 (Transient circuit s
Page 211 and 212: y y 1 y(t) y 0 t t 0 t 1 Fig. 133 e
Page 213 and 214: for discrete evolution defined on I
Page 215 and 216: ⇒ if y ∈ C 2 ([0, T]), then y(t
Page 217 and 218: The implementation of an s-stage ex
Page 219 and 220: Example 11.5.2 (Blow-up). ✸ toler
Page 221 and 222:
However, it would be foolish not to
Page 223 and 224:
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Page 225 and 226:
4 3 2 abstol = 0.000010, reltol = 0
Page 227 and 228:
✸ ✸ Motivated by the considerat
Page 229 and 230:
0 1 2 3 4 5 6 u(t),v(t) 0.01 0.008
Page 231 and 232:
MATLAB-CODE : Explicit integration
Page 233 and 234:
Shorthand notation for Runge-Kutta
Page 235 and 236:
13 Structure Preservation 13.1 Diss
Page 237 and 238:
[18] G. GOLUB AND C. VAN LOAN, Matr
Page 239 and 240:
linear in Gauss-Newton method, 538
Page 241 and 242:
Chebychev nodes, 678 double, 634 fo
Page 243 and 244:
x∗ n y ˆ= discrete periodic conv
Page 245 and 246:
Gaussian elimination with pivoting
show all

Numerical Methods Contents - SAM

Create successful ePaper yourself

Delete template?

Save as template?