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(row transformation = multiplication with elimination matrix) ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 0 · · · · · · 0 a 1 a 1 − a 2 a 1 0 1 a 1 ≠ 0 − a a 2 0 3 a 1 a 3 = 0 . ⎜ ⎝ . ⎟⎜ ⎠⎝ . ⎟ ⎜ ⎠ ⎝ . ⎟ ⎠ − a n a1 0 1 a n 0 n − 1 steps of Gaussian elimination: ➤ matrix factorization (→ Ex. 2.1.1) (non-zero pivot elements assumed) A = L 1 · · · · · L n−1 U with elimination matrices L i , i = 1, ...,n − 1 , upper triangular matrix U ∈ R n,n . ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 0 · · · · · · 0 1 0 · · · · · · 0 1 0 · · · · · · 0 l 2 1 0 0 1 0 l 2 1 0 l 3 0 h 3 1 = l 3 h 3 1 ⎜ ⎝ . ⎟⎜ ⎠⎝. . ⎟ ⎜ ⎠ ⎝ . . ⎟ ⎠ l n 0 1 0 h n 0 1 l n h n 0 1 Ôº ¾º¾ a from (2.1.1) → Ex. 2.1.1) kk L 1 · · · · · L n−1 are normalized lower triangular matrices (entries = multipliers − a ik ✬ { Lemma 2.2.2 (Group of regular diagonal/triangular matrices). { diagonal A,B upper triangular lower triangular ⇒ AB and A −1 ✫ ✬ ✫ ⎛ ⎜ ⎝ 0 ⎞ · ⎟ ⎜ ⎠ ⎝ (assumes that A is regular) ⎛ 0 ⎞ ⎛ = ⎟ ⎜ ⎠ ⎝ diagonal upper triangular lower triangular 0 . ⎞ . ⎟ ⎠ The (forward) Gaussian elimination (without pivoting), for Ax = b, A ∈ R n,n , if possible, is algebraically equivalent to an LU-factorization/LU-decompositionA = LU of A into a normalized lower triangular matrix L and an upper triangular matrix U. ✬ ✩ ✪ ✩ Ôº ¾º¾ ✪ ✩ Definition 2.2.1 (Types of matrices). A matrix A = (a ij ) ∈ R m,n is • diagonal matrix, if a ij = 0 for i ≠ j, Lemma 2.2.3 (Existence of LU-decomposition). The LU-decomposition of A ∈ K n,n exists if and only if all submatrices (A) 1:k,1:k , 1 ≤ k ≤ n, are regular. • upper triangular matrix if a ij = 0 for i > j, ✫ ✪ • lower triangular matrix if a ij = 0 for i < j. A triangular matrix is normalized, if a ii = 1, i = 1,...,min{m, n}. ⎛ ⎞ ⎛ ⎞ ⎛ 0 ⎜ ⎝ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎠ ⎝ ⎠ ⎝ 0 ⎞ ⎟ ⎠ Proof. by induction w.r.t. n, which establishes existence of normalized lower triangular matrix ˜L and regular upper triangular matrix Ũ such that ( ) ( ) ( ) Ã b ˜L 0 Ũ y a H = α x H =: LU . 1 0 ξ Then solve for x,y and ξ ∈ K. Regularity of A involves ξ ≠ 0 so that U will be regular, too. Remark: Uniqueness of LU-decomposition: diagonal matrix upper triangular lower triangular Ôº ¾º¾ Regular upper triangular matrices and normalized lower triangular matrices form matrix groups. Their only common element is the identity matrix. L 1 U 1 = L 2 U 2 ⇒ L −1 2 L 1 = U 2 U −1 1 = I . Ôº ¾º¾
A direct way to LU-decomposition: ➤ ⎛ ⎜ ⎝ LU = A ⇒ a ik = 0 ⎞ ⎛ · ⎟ ⎜ ⎠ ⎝ • row by row computation of U min{i,k} ∑ j=1 • column by column computation of L Entries of A can be replaced with those of L, U ! (so-called in situ/in place computation) ˆ= rows of U ˆ= columns of L (Crout’s algorithm) LU-factorization = “inversion” of matrix multiplication: ⎞ ⎛ = ⎟ ⎜ ⎠ ⎝ ⎧∑ ⎨ i−1 j=1 l ij u jk = l iju jk + 1 · u ik , if i ≤ k , ⎩ ∑ k−1 j=1 l iju jk + l ik u kk , if i > k . 1 2 3 1 2 3 4 ⎞ ⎟ ⎠ Fig. 2 Ôº ¾º¾ Code 2.2.2: LU-factorization 1 function [ L ,U] = l u f a k (A) 2 % LU-factorization of A ∈ K n,n 3 n = size (A, 1 ) ; i f ( size (A, 2 ) ~= n ) , error ( ’ n ~= m ’ ) ; end 4 L = eye ( n ) ; U = zeros ( n , n ) ; 5 for k =1:n 6 for j =k : n 7 U( k , j ) = A( k , j ) − L ( k , 1 : k−1)∗U( 1 : k−1, j ) ; 8 end 9 for i =k +1:n 10 L ( i , k ) = (A( i , k ) − L ( i , 1 : k−1)∗U( 1 : k−1,k ) ) /U( k , k ) ; 11 end 12 end ✛ ✚ Code 2.2.3: matrix multiplication L · U 1 function A = l u m u l t ( L ,U) 2 % Multiplication of normalized lower/upper triangular matrices 3 n = size ( L , 1 ) ; A = zeros ( n , n ) ; 4 i f ( ( size ( L , 2 ) ~= n ) | | ( size (U, 1 ) ~= n ) | | ( size (U, 2 ) ~= n ) ) 5 error ( ’ size mismatch ’ ) ; end 6 for k=n:−1:1 7 for j =k : n 8 A( k , j ) = U( k , j ) + L ( k , 1 : k−1)∗U( 1 : k−1, j ) ; 9 end 10 for i =k +1:n 11 A( i , k ) = L ( i , 1 : k−1)∗U( 1 : k−1,k ) + L ( i , k ) ∗U( k , k ) ; 12 end 13 end Ôº½ ¾º¾ asymptotic complexity of LU-factorization of A ∈ R n,n = 1 3 n3 + O(n 2 ) (2.2.1) Remark 2.2.4 (Recursive LU-factorization). Recursive in situ (in place) LU-decomposition of A ∈ R n,n (without pivoting): L,U stored in place of A: 1 function [ L ,U] = l u r e c d r i v e r (A) 2 A = l u r e c (A) ; 3 % post-processing: extract L and U 4 U = t r i u (A) ; 5 L = t r i l (A,−1) + eye ( size (A) ) ; ✘ ✙ 1 function A = l u r e c (A) 2 % insitu recursive LU-factorization 3 i f ( size (A, 1 ) >1) 4 fac = A( 2 : end , 1 ) /A( 1 , 1 ) ; 5 C = l u r e c (A( 2 : end , 2 : end )−fac∗A( 1 , 2 : end ) ) ; 6 A=[A ( 1 , : ) ; fac ,C ] ; 7 end △ Ôº¼ ¾º¾ Solving a linear system of equations by LU-factorization: Algorithm 2.2.5 (Using LU-factorization to solve a linear system of equations). Ôº¾ 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 and 20: Obviously, left multiplication with
Page 21: ❶: elimination step, ❷: backsub
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
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Page 47 and 48: Evident: symmetry of Ã − bbT a 1
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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
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Page 61 and 62: Definition 3.1.3 (Local and global
Page 63 and 64: Example 3.1.6 (quadratic convergenc
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Page 67 and 68: (x (k) ) k∈N0 Cauchy sequence ➤
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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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