Numerical Analysis MATH 128A
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This 3 page Study Guide was uploaded by Kavon Feest on Thursday October 22, 2015. The Study Guide belongs to MATH 128A at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 48 views. For similar materials see /class/226600/math-128a-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.
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Date Created: 10/22/15
H E0 9 F 5 533 7 00 to H 53 H H Math128B January 287 2005 Jonathan Dorfman Background Material on Norms i Hsz denotes the pnorm of a vector in Rni If no p is indicated7 then p 2 is assumed We de ne the operator pnorm7 of A as def HAHp HAHp supi sup HAIHp 0 llIllp Mfl Medium d f lt y I gt2 MI HIH llyll 0051I7y An easy consequence of this is the 77CauchySchwarzBunyakovsky lnequality77 with equality iff z and y parallel llt 97 I gt l S HIH Hyll Medium lfl R A R1 is linear then there is a unique y E R such that 1 lt y z gt VI 6 Rni Since the y E R depends on 17 we write y yli One should think of y as a row vector which operates on I E R by doting z with y 1 I I yl I I I clearly linear in 1 Easy By l7 it makes sense to ask about the operator norm7 of the linear operator 1 R A Rli By 3 this turns out to be equal to where y is determined by Hard The operator norm de ned in 2 can also be given by the following when p 2 HAN lt AI7ygt sup HIHHyH1 Easy Using 5 and 6 we can prove i Easy HABH S For general A and B equality is not achieved7 eg A B 8 i Easy Using 7 and 8 we can prove llA AH S HAH2 Hard Using 3 and 6 we can prove HA AH 2 Easy 9 and 10 imply HA AH lAH2 H E0 H CA3 H H H 5 H 533 H 7 H 9 H to Easy Using 11 we can prove if A is Hermitian7 then HA2 HAH2 note that A g 3 example in 8 is not Hermitian i For any A R A R we de ne the spectrum of A aA 13 A E Cnl A is an eigenvalue of A7 ie is a root of A s characteristic polynomial i For any A R A R we de ne the spectral radius of A pA if the maximum complex norm of the eignevalues of A 77size77 of 0A Easy am W1 and M MA and MA MA Easy pA S 8 3 is example of strict inequality Homework If A is Hermitian7 then pA Easy 17 implies pAA HA AH i Easy 18 and II imply pAA HAH27 iiei pAA formula for HAM Some more de nitions DEFINING PROPERTY NAME SPECTRUM INTERPRETATION A A AA I real orthogonal 0A C 2 E C I preserves Euclidean length A A AAquot I unitary 0A C 2 E C 1 complex version A A real symmetric 0A C R stretch in perpendicular directions A A Hermitian 0A C R complex version A B B positive 0A C R20 symmetric7 positive stretches A 1 A A2 I 1nvolution 0A C 711 oblique re ection A 1 A A re ection 0A C 711 perpendicular re ection A2 A idempotent 0A C 01 oblique projection A2 A A projection 0A C 01 orthogonal projection A A AAquot norma A i diagonalizable via unitary change of basis Additional facts It Diagonalizable matrices share eigenspaces i they commute i simultaneously diagonalizable 2 Every real matrix is sum of symmetric and skewsymmetric A 7A 3 Every complex matrix is sum of Hermitian and skewHermitian A 7A 4 A matrix is unitarily diagonalizable i its Hermitian and skewHermitian parts commute MATH 128A 3101102 SP2009 MT2 STUDY GUIDE Algorithm Problem Matlab Use Case Natural Cubic Spline 3 4 Interpolation ncspl ine Nonpolyi f many sample points Clamped Cubic Spline 3 4 Interpolation ccspline If f is known at endpoints Spline Evaluation 34 Interpolation spl ineeval n1point rule 41 Differentiation Often with n 2 4 and z 10 ih Richardson Extrapi 42 Convi Accel richdemo Improving loworder easily re ned methods NewtonCotes 4 3 Integration Naive method includes trapezoidal Simpsonls Composite Integration 44 Integration Naive method Romberg Integration 45 Integration romberg Certain functions Adaptive Simpson s 46 Integration quad Reaching a required tolerance Gaussian Quadrature 4 Integration gaussquad Smooth functions Eulerls Method 5 IVP Naive method Taylor Methods 53 IVP If derivatives of f are known RungeKutta Methods 54 IVP rk4 If they arenlt 4th order typically most usefuli Interpolating polynomials and osculating polynomials inclusing Taylor polynomials are crucial to the theory of numerical methods Numerical differentiaton is unstable n lpoint methods incur error olt h f l and N eh from roundoff The Richardson extrapolation process changes if eg all error terms are of even order For multiple integrals use iterated integration For improper integrals use substitutions and power series around vertical asymptotesi An integratorls degree of precision is the degree of polynomial it will always integrate exactlyi Gauss ian quadrature optimizes iti IVP is of the form yt fty a S t S b ft0 yer A Lipschitz constant is usually a bound on for t in the domain and y in the range77 of Details are in Bill If f is continuous and a Lips IVP is wellposed solving a small perturbation of the IVP causes a small error chitz constant exists tual actual An IVPsolver s local truncation error at each node ti is W 7 tiy m alhii Here hi ti1 7 ti and a step of the IVP solver is represented as yi1 M hi ti yi hi Higherorder Taylor methods and RungeKutta methods optimize it given different information
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