Numerical Methods Contents - SAM

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Remark 1.2.4 (Scalings). Recall: rules of matrix multiplication, for all K-matrices A,B,C (of suitable sizes), α, β ∈ K Scaling = multiplication with diagonal matrices (with non-zero diagonal entries): associative: (AB)C = A(BC) , bi-linear: (αA + βB)C = α(AC) + β(BC) , C(αA + βB) = α(CA) + β(CB) , non-commutative: AB ≠ BA in general . multiplication with diagonal matrix from left ➤ row scaling ⎛ ⎞⎛ ⎞ ⎛ ⎞ d 1 0 0 a 11 a 12 ... a 1m d 1 a 11 d 1 a 12 ... d 1 a 1m ⎛ ⎜ 0 d 2 0 ⎟⎜a 21 a 22 a 2m ⎟ ⎝ ... ⎠⎝ . . ⎠ = ⎜d 2 a 21 d 2 a 22 ... d 2 a 2m ⎟ ⎝ . . ⎠ = ⎝ d ⎞ 1(A) 1,: . ⎠ . d 0 0 d n a n1 a n2 ... a nm d n a n1 d n a n2 ... d n a n (A) n,: nm multiplication with diagonal matrix from right ➤ column scaling ⎛ ⎞⎛ ⎞ ⎛ ⎞ a 11 a 12 . .. a 1m d 1 0 0 d 1 a 11 d 2 a 12 . .. d m a 1m ⎜a 21 a 22 a 2m ⎟⎜ 0 d 2 0 ⎟ ⎝ . . ⎠⎝ . .. ⎠ = ⎜d 1 a 21 d 2 a 22 . .. d m a 2m ⎟ ⎝ . . ⎠ a n1 a n2 . .. a nm 0 0 d m d 1 a n1 d 2 a n2 . .. d m a nm ) (d 1 (A) :,1 . . . d m (A) :,m Remark 1.2.6 (Matrix algebra). A vector space (V, K, +, ·), where V is additionally equipped with a bi-linear and associative “multiplication” is called an algebra. Hence, the vector space of square matrices K n,n with matrix multiplication is an algebra with unit element I. Remark 1.2.7 (Block matrix product). △ Example 1.2.5 (Row and column transformations). = . △ Ôº¿ ½º¾ Given matrix dimensions M, N,K ∈ N block sizes 1 ≤ n < N (n ′ := N − n), 1 ≤ m < M (m ′ := M − m), 1 ≤ k < K (k ′ := K − k) assume A 11 ∈ K m,n A 12 ∈ K m,n′ B A 21 ∈ K m′ ,n A 22 ∈ K m′ ,n ′ , 11 ∈ K n,k B 12 ∈ K n,k′ B 21 ∈ K n′ ,k B 22 ∈ K n′ ,k ′ . ( ) ( ) ( ) A11 A 12 B11 B 12 A11 B = 11 + A 12 B 21 A 11 B 12 + A 12 B 22 . (1.2.3) A 21 A 22 B 21 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 Ôº¿ ½º¾ Given A ∈ K n,m obtain B by adding row (A) j,: to row (A) j+1,: , 1 ≤ j < n ⎛ ⎞ 1 . . . B = ⎜ ⎝ 1 1 1 . .. A . ⎟ ⎠ 1 M m m ′ n N = m M m ′ left-multiplication right-multiplication with transformation matrices ➙ row transformations column transformations n N n ′ k k ′ n ′ k K k ′ ✸ K △ Ôº¿ ½º¾ Ôº¼ ½º¿
1.3 Complexity/computational effort complexity/computational effort of an algorithm :⇔ number of elementary operators additions/multiplications Crucial: dependence of (worst case) complexity of an algorithm on (integer) problem size parameters (worst case ↔ maximum for all possible data) Usually studied: asymptotic complexity ˆ= “leading order term” of complexity w.r.t large problem size parameters The usual choice of problem size parameters in numerical linear algebra is the number of independent real variables needed to describe the input data (vector length, matrix sizes). Q 2 = A 11 ∗ (B 12 − B 22 ) , Q 3 = A 22 ∗ (−B 11 + B 21 ) , Q 4 = (A 11 + A 12 ) ∗ B 22 , Q 5 = (−A 11 + A 21 ) ∗ (B 11 + B 12 ) , Q 6 = (A 12 − A 22 ) ∗ (B 21 + B 22 ) . Beside a considerable number of matrix additions ( computational effort O(n 2 ) ) it takes only 7 multiplications of matrices of size n/2 to compute C! Strassen’s algorithm boils down to the recursive application of these formulas for n = 2 k , k ∈ N. A refined algorithm of this type can achieve complexity O(n 2.36 ), see [7]. △ Example 1.3.2 (Efficient associative matrix multiplication). operation description #mul/div #add/sub asymp. complexity dot product (x ∈ R n ,y ∈ R n ) ↦→ x H y n n − 1 O(n) tensor product (x ∈ R m ,y ∈ R n ) ↦→ xy H nm 0 O(mn) Ôº½ ½º¿ matrix product (∗) (A ∈ R m,n ,B ∈ R n,k ) ↦→ AB mnk mk(n − 1) O(mnk) ✎ notation (“Landau-O”): f(n) = O(g(n)) ⇔ ∃C > 0, N > 0: |f(n)| ≤ Cg(n) for all n > N. Remark 1.3.1 (“Fast” matrix multiplication). (∗): O(mnk) complexity bound applies to “straightforward” matrix multiplication according to (1.2.1). For m = n = k there are (sophisticated) variants with better asymptotic complexity, e.g., the divideand-conquer Strassen algorithm [39] with asymptotic complexity O(n log 2 7 ): Start from A,B ∈ K n,n with n = 2l, l ∈ N. The idea relies on the block matrix product ( (1.2.3) with A ij ,B ij ∈ K l,l , i,j ∈ {1, 2}. Let C := AB be partiotioned accordingly: C = C11 C 22 C 21 C ). Then 22 tedious elementary computations reveal a ∈ K m , b ∈ K n , x ∈ K n : ( y = ab T) x . average runtime (s) T = a*b’; y = T*x; ➤ complexity O(mn) tic−toc timing, mininum over 10 runs slow evaluation efficient evaluation O(n) 10 0 O(n 2 ) 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 10 0 10 1 10 2 10 3 10 4 problem size n Fig. 1 10 1 Ôº¿ ½º¿ ➤ complexity O(n + m) (“linear complexity”) ( ) y = a b T x . t = b’*x; y = a*t; ✁ average runtimes for efficient/inefficient matrix×vector multiplication with rank-1 matrices ( MATLABtic-toc timing) Platform: MATLAB7.4.0.336 (R2007a) Genuine Intel(R) CPU T2500 @ 2.00GHz Linux 2.6.16.27-0.9-smp C 11 = Q 0 + Q 3 − Q 4 + Q 6 , C 21 = Q 1 + Q 3 , C 12 = Q 2 + Q 4 , C 22 = Q 0 + Q 2 − Q 1 + Q 5 , where the Q k ∈ K l,l , k = 1,...,7 are obtained from Q 0 = (A 11 + A 22 ) ∗ (B 11 + B 22 ) , Q 1 = (A 21 + A 22 ) ∗ B 11 , Ôº¾ ½º¿ Code 1.3.3: MATLAB code for Ex. 1.3.2 1 function d o t t e n s t i m i n g (N, nruns ) 2 % This function compares the runtimes for the 3 % multiplication of a vector with a rank-1 matrix ab T , a,b ∈ R n 4 % using different associative evaluations 5 6 i f ( nargin < 1) , N = 2 . ^ ( 2 : 1 3 ) ; end Ôº ½º¿ 7 i f ( nargin < 2) , nruns = 10; end 8 9 times = [ ] ; % matrix for storing recorded runtimes
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: Remark 1.2.1 (Row-wise & column-wis
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
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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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Definition 3.1.3 (Local and global
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Example 3.1.6 (quadratic convergenc
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k |x (k) − π| L 1−L |x (k) −
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(x (k) ) k∈N0 Cauchy sequence ➤
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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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