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Problem Set 1 1. (25 points) Permuted LU. Consider the lin...

Problem Set 1 1. (25 points) Permuted LU. Consider the linear system Ax = b. In Matlab, implement the permuted LU factorization algorithm which overwrites A with its LU factors (discussed in lecture). Next, implement triangular solvers (backward and forward sub- stitution) which could be used to s...

heat transfer to mercury-filled tube A mercury-filled tube ...

heat transfer to mercury-filled tube A mercury-filled tube (see figure at left) is moved suddenly from ambient conditions, T = 20◦C, into cold air at Tair = −10◦C. The heat transfer coefficient is reasonably constant at h = 9.7 W/m2K (the mercury is a good conductor of heat, and may be ...

A18 (estimation of the remainder for exp) Let f: R R be gi...

A18 (estimation of the remainder for exp) Let f: R R be given through f (x) = exp(x). a) Find the Taylor polynomial Tn,xx (h) and the related remainder Rn,x* (h) of f for the expansion point Xx = 0. b) Give a bound for the remainder R20,0(h) I for X E [-5,5]. A19 (Taylor polynomial of the i...

#6 (40 points). In this problem, you will implement a code f...

#6 (40 points). In this problem, you will implement a code for the natural cubic spline interpolation of the data obtained from 1 f(x) = defined on [-1,1]. - 1 + 25 x² Let n be a positive integer and consider uniform nodal points x. = i.h - 1, h = 2/n (0 < i < n) a. Produ...

1. y′ =t+y, y(0)=0 0≤t≤4 Compare the true sol...

1. y′ =t+y, y(0)=0 0≤t≤4 Compare the true solution with the approximate solutions from t = 0 to t = 4, with the step size h = 0.5, obtained by each of the following methods. (Note : True Solution is : y(t) = et − t − 1). a. Heun’s method b. Midpoint method Pre...

The SIR model of epidemiology models the spread of an infect...

The SIR model of epidemiology models the spread of an infectious pathogen. Divide a population into three portions: the susceptible (S), the infectious (1). and the recovered (R). Here S. 1, and R are fractions of a total population P with S I+R 1. The fraction in each subpopulation evolves thro...

Householder Transformation: To transform a given symmetric m...

Householder Transformation: To transform a given symmetric matrix A to Equivalent Transformation tridiagonal form the transformation = /-2ww' called the Householder an In fact the matrix is applied. Here weR" and w'w=1 P is also symmetric and orthogonal. That means P=P" Comment...

Problem 1. (35 pts.) Show that for the matrix A= -1 5 2 2 ...

Problem 1. (35 pts.) Show that for the matrix A= -1 5 2 2 , the Gauss-Seidel method generally fails, i.e., it does not converge for arbitrary initial vector x(0) € R². (The argument that matria A is not a strictly diagonally dominant does not work, since this is only a sufficient con...

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