Final Exam ReviewTopics Included • Dense Linear Systems – Gaussian Elimination – LU Decomposition – Matrix and Vector Norms – Condition Number • Matrix Factorizations – Symmetric Positive Definite - Cholesky – Linear Least Squares – Orthogonal Matrices adn QR Factorization ∗ Householder transformation ∗ Givens rotation ∗ Gram-Schmidt • Eigenvalue Problem – Computation of Eigenvalue Eigenvector pairs by hand – Similarity Transformation – QR Algorithm for Computation of Eigenvalues (proof not required) – Power iteration for Eigenvector computation – Inverse iteration for Eigenvector computation • Nonlinear Root Finding – Convergence rate computation – Bisection method – Regula Falsi – Secant Method – Fixed point iterations – Newton’s method - scalar valued functions • Polynomial Interpolation – Computation of interpolation for given basis - e.g., Monomial basis – Newton interpolation – Lagrange interpolation – Piecewise interpolation – Spline interpolation • Numerical Differentiation and Integration – Error bounds with Taylor series – Computation of approximate derivatives with Taylor series – Hand calculation with Richardson extrapolation – Trapezoid rule – Quadrature through method of undetermined coefficients – Gaussian quadrature hand calculation Page 1 of 7 for conceptual material. please refer books such as Heath.approximately 75 % will consist of numericals with the rest being conceptual questions. Watkins etc. Trapezoid. Adams methods • Iterative Methods for Linear Systems – Splitting a matrix – Jacobi/Gauss-Siedel – Sucessive Over Relaxation Methods – Steepest Descent Method Practise Problems This exam will focus on numericals . ∗ Find angles α. Heun – Methods : Backward Euler. RK methods. and γ such that 2sinα − cos β + 3tanγ = 3 4sinα + 2cos β − 2tanγ = 2 6sinα − 3cos β + tanγ = 9 where 0 ≤ α ≤ 2π and 0 ≤ β ≤ 2π and 0 ≤ γ ≤ π . Page 2 of 7 . – Dense Linear Solve ∗ Solve the following system with GE and back substitution: v−w = 3 −2u + 4v − w = 1 −2u + 5v − 4w = −2 ∗ Apply GE with back substitution: 2 x1 − x2 = 0 − x1 + 2 x2 − x3 = 0 − x2 + x3 = 1 ∗ GE with back substitution: 4 x2 − 3 x3 = 3 − x1 + 7 x2 − 5 x3 = 4 − x1 + 8 x2 − 6 x3 = 5 ∗ Consider the following system: 4x − 8y + 5z = b1 4x − 7y + 4z = b2 3x − 4y + 2z = b3 Solve for ba = [100] T and bb = [010] T and bc = [001] T at one time be performing GE on an augmented matrix of the form [ A|ba |bb |bc ]. This review will hence focus on numericals. β.Final Exam Review • Differential Equations – Stability of system of equations – Error analysis for numerical schemes – Order of accuracy of methods – Methods : Forward Euler. 5469 0.9058 0.9649 0.9575 0.9572 0.4854 0.1270 0.3 = 0 by a rotation of rows 2 and 3 when A = 0 0 5 0 0 12 −3 0 1 1 ∗ Use Givens rotation to find the QR decomposition of A = 1 2 3 1 1 1 0. 3 2 6 7 1 ∗ Use Givens rotation to set a3.9706 ∗ Use Givens rotation to compute the QR factorization of A = 0.6324 0.0975 0.1576 0.9134 0.5 3 6] T . 1 1 1 1 2 4 ∗ Compute the QR factorization by Householder transformation: 1 3 9 1 4 16 3 −2 3 x1 ∗ Compute least squares solution by Householder transformation: 0 3 = 5 x2 4 4 4 1 1 1 1 2 4 3 and y = [1 1.Final Exam Review ∗ Attempt to solve the sytem with GE: − x1 + 3 x2 − 2 x3 = 1 − x1 + 4 x2 − 3 x3 = 0 − x1 + 5 x2 − 4 x3 = 0 2 −2 ∗ Perform an LU decomposition on the following matrix: 1 −3 3 7 2 −1 2 9 ∗ LU the following matrix: 4 1 8 5 24 1 2 3 ∗ LU decomposition for rank deficient matrix: 2 4 6 7 1 9 – Matrix Factorizations 4 1 5 1 2 1 ∗ Find the Cholesky decomposition of the following symmetric matrix: 2 8 4 1 4 11 1 2 2 ∗ Find the Cholesky decomposition of: 2 8 0 2 0 24 1 −1 1 2 x 1 −1 1 1 1 x2 = ∗ Use Cholesky factorization to find the least squares solution: 0 1 1 −1 x3 0 1 −1 −2 −1 1 1 1 x 1 1 −1 1 1 x2 = ∗ Use Cholesky factorization to find the solution to the least squares problem: 1 0 1 −1 x3 0 1 −1 −1 T T ∗ Use Householder transformation to transform a = [1 2 2] to [c 0 0] . ∗ Minimize c ∈ R in || Ac − y||with: A = 1 3 9 1 4 16 ∗ Zero the second element of x = [11/2] T using Givens rotation.2785 0.8147 0.8003 Page 3 of 7 . Provide a satisfactory explanation for this slowness. x2 . ∗ Rayleigh Quotient: If x is an eigenvector of a matrix A. 1 2 ∗ Use Gram-Schmidt to find the QR factorization for A = 1 2 0 3 ∗ Use Gram-Schmidt to find an orthonormal basis for the subspace in R4 spanned by the following vectors: [1 1 1 1] T . but does have two linearly independent eigenvectors. 4 4 – Eigenvalues and Eigenvectors 1 2 ∗ What is the characteristic polynomial of 3 4 1 1 1 ∗ What is the characteristic polynomial of 2 3 4 0 1 1 1 3 ∗ What are the eigenvalues of 0 3 ∗ Find a 2 × 2 matrix that has only the eigenvalue zero.1 and b = 1. Find them correct to two signnificant digits with Newton’s method. −4 10 ∗ Predict the rate of convergence(qualitatively) of the power method for A = 7 5 – Nonlinear Root Finding ∗ Calculate the dominant eigenvalue of A = ∗ Find a root of f ( x ) = cos( x ) − cos(3x ) using the bisection method. x3 } with x1 = [1 2 3 0] T and x2 = [1 2 0 0 ] T and x3 = [1 0 0 1] T is a subspace of R4 . then the corresponding eigenvalue is given by λ= Ax · x x·x Use the results from the previous example to approximate the dominant eigenvalue of A. 0 −1 1 1 ∗ Find eigenvalues and eigenvectors for the following matrices: and 2 3 0 1 0 1 1 ∗ Find eigenvalues and eigenvectors of the following matrix: 1 0 1 1 1 0 0 1 ∗ Find eigenvalues and eigenvectors of −1 0 ∗ When does the QR algorithm converge to a diagonal matrix? ∗ Why do shifts help in convergence of the QR algorithm? 2 −12 ∗ Compute six iterations of the power method to approximate the dominant eigenvector of A = 1 −5 with an initial approximation of x0 = [1 1] T .Final Exam Review ∗ Suppose W = span{ x1 . Look up reference books to 6 5 determine the rate of convergence of the power method. Find an orthogonal basis for W . Page 4 of 7 . Use Gram-Schmidt to find and orthogonal basis for W . ∗ Suppose W = span{ x1 . 4 5 to three decimal places. ∗ Using a calculator. x2 } where x1 = [1 1 0] T and x2 = [2 2 3] T . with x0 = 1.0 how many steps of the bisection methods are needed to determine the root with an −8 error of at most 1 2 × 10 ? ∗ Two of the four zeroes of f ( x ) = x4 + 2x3 − 7x2 + 3 are positive. ∗ If a = 0.1. observe the slowness of Newton’s method for f ( x ) = ( x − 1)8 . ∗ Find a root of f ( x ) = x4 − 4x3 + 12 to withing 1e-2 with a = 1 and b = 2. 3 −2 ∗ Use Gram-Schmidt to find the QR factorization of A = 0 3 . [−1 4 4 1] T and [4 − 2 2 0] T . ∗ Find the natural cubic spline that interpolates the data: {(1. 3]) – Interpolation ∗ Find an interpolating polynomial for x = [1.(1. 1. what will x1 be? If Newton’s method is used on f ( x ) = x5 − x3 + 3 and if xn = 1. 2. 4/3. 2). (3. to what values? If Newton’s method is used on f ( x ) = x3 − x + 1 and x0 = 1. 9. – Numerical Differentiation and Integration ∗ Using Taylor series.0). 3/32. 4] and y = [−1. (This can be done with little work). what is xn+1 . 1. (4. −1. By adding one additional term to p.1). establish the error term for the formula f (0) ≈ ∗ Derive the approximation formula f (x) ≈ 1 [4 f ( x + h) − 3 f ( x ) − f ( x + 2h)] 2h 1 [ f (2h) − f (0)] 2h 1 2 and show that its error term is of the form 3 h f ( ) ∗ Derive the following two formulas f (x) ≈ f (x) ≈ 1 [ f ( x + 2h) − f ( x − 2h)] 4h 1 [ f ( x + 2h) − 2 f ( x ) + f ( x − 2h)] 4 h2 Page 5 of 7 . 0. 1.5. −1. 61].2). 259]. ∗ The polynomial p( x ) = x4 − x3 + x2 − x + 1 has the following values:x = [−2. (2.(0. 11.0). 3] and q( x ) = [31.Final Exam Review ∗ Using the Newton iteration defined by x n +1 = 1 2 xn + R xn what does xn converge to? ∗ What is Newton’s method for determining the reciprocal of the square root of a positive number? Perform √ two iterations to approximate 1/ 5 with x0 = 1 and x0 = −1. use Newton interpolating polynomial and find an approximation for f (4. (All roots are in [−3. 1. 1/2). 93. 4. Newton’s method. ∗ Given the data x = [0. For what starting values will Newton’s method converge for the function x2 /(1 + x2 )? Calculate an approximate value of 43/4 using one step of the secant method with x0 = 3 and x1 = 2. 0. 23. 1). what is x2 ? If the secant method is used on f ( x ) = x5 + x3 + 3 and if xn−2 = 0 and xn−1 = 1 what is xn ? Using the bisection method. ∗ Consider the following separate procedures: ( a ) x n +1 = ( b ) x n +1 1 r 2 xn − 2 3 xn 1 1 = xn + 2 xn ∗ ∗ ∗ ∗ ∗ ∗ ∗ Do they converge for any nonzero initial point? If so. 2. Find a polynomial q( x ) that takes these values: x = [−2. 6] and y = [1. 1). 1/3). If we use the secant method on f ( x ) = x3 − 2 x + 2 starting with x0 = 0 and x1 = 1. 25]. ∗ The polynomial p( x ) = 2 − ( x + 1) + x ( x + 1) − 2x ( x + 1)( x − 1) interpolates the first four points in the table (-1. and the secant method. (1.(3. 11. 5. (1/2. 1. 1/4)}. 2). ∗ Find the periodic cubic spline that interpolates the data: (0.-7). (2. 30]. 2. 10). 5. 3] with p( x ) = [31. find the largest positive root correct to three decimal places of x3 − 5 x + 3 = 0. 3. 2. 1. find the polynomial that interpolates the whole table. 1. −1/3. 2. ∗ Consider 2 1 1 dx / x3 . Interpolate the function with a natural cubic spline and approximate the integral 2dx 11 =π 1 + x2 − . (e) Is Forward Euler stable for this stepsize? (f) Is Backward Euler stable for this stepsize? ∗ For the IVP x = x2 et + x with x (0) = 1 and x (0) = 2 calculate an approximate value for x (0.5 using Backward Euler. 4. (b) What are the corresponding initial conditions for the system variables? (c) Is the system stable? (d) Perform one step of Forward Euler with h = 0. 1] with a stepsize h = 0. ∗ Consider the IVP y = y for t ≥ 0 with y(0) = 1 and y (0) = 2. Compare the result with Simpson’s rule. Use backward Euler to compute the solution at t = 0. f (x) = – Ordinary Differential Equations ∗ Is the following system of ODEs stable? y1 = − y1 + y2 y2 = −2y2 ∗ Consider the ODE y = −5y with y(0) = 1. Apply the trapezoid rule for n = 1. 3/2. (5/4. What is the result of using the trapezoid rule with partition points 1. Compare your answers with a MATLAB simulation of this ODE. compute the approximate solution for y at t = 1 using the (a) Forward Euler method and (b) Backward Euler method.5. 1.6). yi ) data: (1.01 to compute 2 sin( x )dx. Is this ODE stable? With a stepsize h = 0. ∗ Solve the IVP x = −tx2 with x (0) = 2 at t = −0.1. 4 ∗ Perform an analysis similar to the previous problem. n unequal intervals. Compare your answers.1) using one step of the Taylor series method. (a) Express this as an equivalent first order system.1).7). Take one steo of the Tayloer series method of order 2 with h = 0.Final Exam Review ∗ Consider the integral 0 e x dx. (7/4.1 and Runge-Kutta method of order 2 to recompute x (1. ∗ Use Heun’s method to solve the IVP y = −y + t + 1/2 with y(0) = 1 in the interval [0. (2.5 with the Forward Euler method. ∗ Consider the IVP y = −y2 with y(0) = 1. Divide the interval [0. ∗ Consider the IVP x = (tx )3 − ( x /t)2 with x (1) = 1.5000. by taking 201 equally spaced points for 0 cos( x3 )dx ∗ Approximate x f ( x )dx for the following ( xi . 1] into n uniform partitions.5.1. and n uniform intervals respectively: a+h a b a b a 2 f ( x )dx ≈ h f n −1 1 a+ h 2 1 xi + hi 2 f ( x )dx ≈ i =0 ∑ hi f n −1 i =0 f ( x )dx ≈ h ∑ 1 f a + (i + ) h 2 ∗ Suppose that the function 2 1 + x2 is known only at x1 = −1. is Forward Euler stable? Compute the numerical solution at t = 0. ∗ With an inital value of y(0) = 1 at t = 0 and a stepsize of h = 1. Page 6 of 7 . and 2? 5 ∗ If the trapezoid rule is used with h = 0. x2 = 0 and x3 = 1.5) ∗ Obtain the following expressions for the midpoint rule error terms for one interval.8). Use Forward Euler as a predictive guess for the Newton iteration (predictor) and the nonlinear solution of backward Euler with this initial guess is the corrector. Is Backward Euler stable for this ODE and stepsize? Compute the solution for t = 0.1 with stepsize h = 0. 2. Does the error converge as n increases? ∗ Repeat the previous problem with Simpson’s rule. what numerical value will the absolute error not exceed? Give the best answer based on the error formula.0).2 using one step of the second order Runge Kutta method. −2. ∗ Use Gauss Siedel for solving 10x1 − x2 + 2x3 − x1 + 11x2 − x3 + 3x4 2x1 − x2 + 10x3 − x4 3 x2 − x3 + 8 x4 starting with [0 0 0 0] T and performing three iterations.Final Exam Review – Iterative Solutions for Sparse Systems ∗ Let 3 1 A= 2 4 −1 1 1 −1 3 and b = [2. ∗ Consider the linear system =6 = 25 = −11 = 15 9x1 + x2 + x3 = b1 2x1 + 10x2 + 3x3 = b2 3x1 + 4x2 + 11x3 = b3 Starting with x = [0 0 0] T . Compare your results with the true solution of the system. Page 7 of 7 . ∗ Apply the Jacobi method to solve 5 x1 − 2 x2 + 3 x3 = −1 −3 x1 + 9 x2 + x3 = 2 2 x1 − x2 − 7 x3 = 3 with initial guess [0. 0.1. compare the result after three iterations of the Jacobi and Gauss-Siedel methods for (a) b = [16 44 59] T and (b) b = 0 − 7 7] T ∗ Consider the linear system: 10x1 + x2 = 12 x1 + 10x2 = 21 Starting with x = [0 0] T . − 1. 0] T . 0] T upto three decimal points. apply the Jacobi iteration to get three digits of the solution. −5] T . Will Jacobi iterations converge for the solution of Ax = b? If so. apply Gauss-Siedel for upto three iterations and the Successive Over Relaxation method for three iterations with ω = 1. starting with [1.