Numerical Methods Contents - SAM

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⎧⎪ ⎨ a ij − a ik a a ′ ij := a kj for k < i,j ≤ n , kk 0 for k < i ≤ n,j = k , ⎪ ⎩ a ij else, ⎧ ⎨ b ′ i := b i − a ik b a k for k < i ≤ n , kk ⎩ b i else. (2.1.1) ★ ✧ asymptotic complexity (→ Sect. 1.3) of Gaussian elimination (without pivoting) for generic LSE Ax = b, A ∈ R n,n = 2 3 n3 + O(n 2 ) ✥ ✦ multipliers l ik Remark 2.1.3 (Gaussian elimination via rank-1 modifications). Block perspective (first step of Gaussian elimination with pivot α ≠ 0), cf. (2.1.1): ⎛ α c T ⎞ ⎛ α c T A := d C → A ′ := 0 C ′ := C − dcT α ⎜ ⎟ ⎜ ⎝ ⎠ ⎝ ⎞ . (2.1.3) ⎟ ⎠ Algorithm 2.1.2. C++-code snippet: In place (in-situ) implementation of Gaussian elimination for LSE Ax = b Never implement Gaussian elimination yourself ! use numerical libraries (LAPACK) or MATLAB ! MATLAB operator: \ template linsolve(Matrix &A,Vector &b) { int n = A.dim(); for(int i=1;i
❶: elimination step, ❷: backsubstitution step 2.2 LU-Decomposition/LU-Factorization ⎛ ⎛ Assumption: (sub-)matrices regular, if required. Remark 2.1.5 (Gaussian elimination for non-square matrices). “fat matrix”: A ∈ K n,m , m>n: ⎛ ⎞ ⎞ ⎜ ⎟ ⎝ ⎠ −→ ⎜ ⎟ ⎝ 0 ⎠ −→ ⎜ ⎝ elimination step back substitution 1 1 0 0 1 ⎞ ⎟ ⎠ △ The gist of Gaussian elimination: ⎛ ⎞ ⎛ ⎞ ⎛ ⎜ ⎟ ⎝ ⎠ −→ ⎜ ⎟ ⎝ 0 ⎠ −→ ⎜ ⎝ 1 1 0 row transformations row transformations row transformation = adding a multiple of a matrix row to another row, or multiplying a row with a non-zero scalar (number) (ger.: Zeilenumformung) Note: row transformations preserve regularity of a matrix (why ?) 0 1 ⎞ ⎟ ⎠ Simultaneous solving of LSE with multiple right hand sides Given regular A ∈ K n,n , B ∈ K n,m , seek X ∈ K n,m MATLAB: AX = B ⇔ X = A\B; X = A −1 B asymptotic complexity: O(n 2 m) Code 2.1.6: Gaussian elimination with multiple r.h.s. 1 function X = gausselim (A, B) 2 % Gauss elimination without pivoting, X = A\B 3 n = size (A, 1 ) ; m = n + size (B, 2 ) ; A = [ A,B ] ; 4 for i =1:n−1, p i v o t = A( i , i ) ; 5 for k= i +1:n , fac = A( k , i ) / p i v o t ; 6 A( k , i +1:m) = A( k , i +1:m) − fac∗A( i , i +1:m) ; 7 end 8 end 9 A( n , n+1:m) = A( n , n+1:m) / A( n , n ) ; 10 for i =n−1:−1:1 11 for l = i +1:n 12 A( i , n+1:m) = A( i , n+1:m) − A( l , n+1:m) ∗A( i , l ) ; 13 end 14 A( i , n+1:m) = A( i , n+1:m) /A( i , i ) ; 15 end 16 X = A ( : , n+1:m) ; △ Ôº½ ¾º½ A matrix factorization (ger. Matrixzerlegung) writes a general matrix A as product of two special (factor) matrices. Requirements for these special matrices define the matrix factorization. Mathematical issue: existence & uniqueness Ôº¾ ¾º¾ <strong>Numerical</strong> issue: algorithm for computing factor matrices Ôº¿ ¾º¾ Example 2.2.1 (Gaussian elimination and LU-factorization). LSE from Ex. 2.1.1: consider (forward) Gaussian elimination: ⎛ ⎝ 1 1 0 ⎞ ⎛ 2 1 −1⎠ ⎝ x ⎞ ⎛ 1 x 2 ⎠ = ⎝ 4 ⎞ x 1 + x 2 = 4 1 ⎠ ←→ 2x 1 + x 2 − x 3 = 1 . 3 −1 −1 x 3 −3 3x 1 − x 2 − x 3 = −3 ⎛ ⎝ 1 ⎞ ⎛ 1 ⎠ ⎝ 1 1 0 ⎞ ⎛ 2 1 −1 ⎠ ⎝ 4 ⎞ ⎛ 1 ⎠ ➤ ⎝ 1 ⎞ ⎛ 2 1 ⎠ ⎝ 1 1 0 ⎞ ⎛ 0 −1 −1 ⎠ ⎝ 4 ⎞ −7 ⎠ ➤ 1 3 −1 −1 −3 0 1 3 −1 −1 −3 ⎛ ⎞ ⎝ 1 2 1 ⎠ 3 0 1 ⎛ ⎝ 1 1 0 ⎞ 0 −1 −1 ⎠ 0 −4 −1 ⎛ ⎝ 4 −7 −15 ⎞ ⎠ ➤ ⎛ ⎞ ⎝ 1 2 1 ⎠ 3 4 1 } {{ } =L = pivot row, pivot element bold, negative multipliers red ⎛ ⎞ ⎝ 1 1 0 0 −1 −1 ⎠ } 0 0 {{ 3 } =U Perspective: link Gaussian elimination to matrix factorization → Ex. 2.2.1 ⎛ ⎝ 4 −7 13 ⎞ ⎠ ✸ Ôº ¾º¾
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: Obviously, left multiplication with
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
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Given x (k) ∈ I, next iterate :=
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secant method ( MATLAB implementati
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Assuming p = 1: p > 1: ∥ C ∥e (
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This is a simple computation: DG(x)
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k x (k) ǫ k := ‖x ∗ − x (k)
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Code 3.4.14: Damped Newton method (
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MATLAB-CODE: Broyden method (3.4.11
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Algorithm 4.1.3 (Steepest descent).
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Example 4.1.8 (Convergence of gradi
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4.2.1 Krylov spaces Definition 4.2.
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Remark 4.2.3 (A posteriori terminat
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10 figure ; view ([ −45 ,28]) ; m
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Idea: Solve Ax = b approximately in
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eplaced with κ(A) ! 4.4.2 Iteratio
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For circuit of Fig. 55 at angular f
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(Linear) generalized eigenvalue pro
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10 0 10 1 matrix size n d = eig(A)
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0 50 100 150 200 250 300 350 400 45
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1 2 3 k ρ (k) EV ρ (k) EW ρ (k)
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✬ ✩ ✬ ✩ Lemma 5.3.4 (Ncut a
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In other words, roundoff errors may
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Theory: linear convergence of (5.3.
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error in eigenvalue 10 0 10 −2 10
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✬ Residuals r 0 ,...,r m−1 gene
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Algebraic view of the Arnoldi proce
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5.5 Singular Value Decomposition Re
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Illustration: columns = ONB of Im(A
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✬ Theorem 5.5.7 (best low rank ap
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Reassuring: Remark 6.0.4 (Pseudoinv
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Consider the linear least squares p
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Goal: Euclidean distance of y ∈ R
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6.5 Non-linear Least Squares If (6.
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0 2 4 6 8 10 12 14 16 value of ∥
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Definition 7.1.1 (Discrete convolut
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Expand a 0 ,...,a n−1 and b 0 , .
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(7.2.2) is a simple consequence of
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Dominant coefficients of a signal a
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11 c = f f t ( y ) ; 12 13 figure (
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Two-dimensional trigonometric basis
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8 end 9 t1 = min ( t1 , toc ) ; 10
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Step II: for k =: rq + s, 0 ≤ r <
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MATLAB-CODE Sine transform function
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△ Example 7.5.2 (Linear regressio
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[23, Ch. IX] presents the topic fro
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Code 8.1.3: Horner scheme, polynomi
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1.2 equality in (8.2.10) for y := (
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ecursive definition: p i (t) ≡ y
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a 1 = y 1 − a 0 t 1 − t 0 = y 1
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Observations: Strong oscillations o
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−1 −0.8 −0.6 −0.4 −0.2 0
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8.5.3 Chebychev interpolation: comp
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9.1 Shape preserving interpolation
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9.2.2 Piecewise polynomial interpol
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Interpolation of the function: f(x)
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2 % Plot convergence of approximati
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9.4 Splines Definition 9.4.1 (Splin
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➤ Linear system of equations with
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y i+1 t i−1 t i t i+1 y i−1 y i
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35 h= d i f f ( t ) ; 36 d e l t a
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1 0.9 Function f 1 0.9 Function f
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f 10 Numerical Quadrature Numerical
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f • n = 1: Trapezoidal rule • n
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For fixed local n-point quadrature
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Equidistant trapezoidal rule (order
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Heuristics: A quadrature formula ha
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20 Zeros of Legendre polynomials in
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|quadrature error| 10 0 Numerical q
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f f • line 9: estimate for global
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Model: autonomous Lotka-Volterra OD
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Example 11.1.6 (Transient circuit s
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y y 1 y(t) y 0 t t 0 t 1 Fig. 133 e
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for discrete evolution defined on I
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⇒ if y ∈ C 2 ([0, T]), then y(t
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The implementation of an s-stage ex
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Example 11.5.2 (Blow-up). ✸ toler
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However, it would be foolish not to
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
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4 3 2 abstol = 0.000010, reltol = 0
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✸ ✸ Motivated by the considerat
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0 1 2 3 4 5 6 u(t),v(t) 0.01 0.008
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MATLAB-CODE : Explicit integration
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Shorthand notation for Runge-Kutta
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13 Structure Preservation 13.1 Diss
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[18] G. GOLUB AND C. VAN LOAN, Matr
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linear in Gauss-Newton method, 538
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Chebychev nodes, 678 double, 634 fo
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x∗ n y ˆ= discrete periodic conv
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Gaussian elimination with pivoting
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