(a) Use three-digit-chopping at each step, and Gaussian Elimination without pivoting, to solve the linear system,...

   

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(a) Use three-digit-chopping at each step, and Gaussian Elimination without pivoting, to solve the linear system, 0.011x+59.9y= 58.1; 5.44x-7.33y=48.1 (b) Use three-digit-rounding and Gaussian Elimination "with pivoting" to solve the same problem in part(a). Show the main steps in your Gaussian Elimination. #2(20 points). (a) If (I - A) is invertible and A=0, prove that (IA)=I+ A+ A. (b) Prove that the product of two orthogonal matrices A and B will also be orthogonal. (c) For the matrix A = [Cose Sine Sine], Compute A37, the thirty-seventh power of A, without -Cose actually multiplying out 37 times. Give reasons and show your steps. Do not use a calculator. (d) For some matrix C=[mm], find the largest singular value. Given m>0". [114] #3(15 points). The matrix A= 1 4 1 Obtain an LU Decomposition for this matrix. Do not use 411 decimals at all. Show your steps. Write the matrix L and U. #4(15 points). While solving the system, Ax=b, we came up with a formula for the relative error ||x-x|| as, ||x|| K(A), where x is an approximate solution, K(A) is the condition number and ||b||' r is the corresponding residual. (a) Prove this formula in your own words. Do not copy the proof from our book. (b) Explain the importance of this formula. #5(15 points). Fit a "least squares function P(x)" of the form P(x)= ACosx + Bx for the data shown below. If needed, round your final answers to 3 decimal digits. Show your normal equations and calculations clearly. x 0 2 y 1 3 10 #6(20 points). (a) Jacobi's method for solving a system Ax=b can be written in iterative form as: xk+1 = Txk+c. Suppose the method converges to the vector x, such that x=Tx+c. Prove that ||xn-x|| SITI (Recall that as n approaches infinity, Xn-x is the final error vector and xo-x is the ||Xox|| initial error vector). You can take any norm, L-2 or L-infinity. (b) Suppose the initial error vector is (xo-x)= [1 -2 2 0]T and the matrix T is symmetric with largest eigenvalue = 0.88. Use L norm to estimate the number of iterations required for Jacobi method to converge within 106 of the exact answer. #7(15 points). Find the "continuous" least squares polynomial of the form P(x) = A + Bx for the function f(x) = Cosx on the interval [0,]. Use two decimal digits in your final answer. For reference, the integral of xCosx is xSinx+Cosx. 3 4 2 #8(15 points). It is known that f(x) = x- x++... Compute a Pade' approximation for f(x), in the form: a+bx 1+cx+dx2' 4 -1 -1 0 4 0 #9(20 points). Suppose A = ,x= b= , xo-initial vector = 1 0 4 0 -1-1 4 1001 (a) Apply Gauss Seidel method to obtain vectors X1 and X2 in solving the system Ax=b. You must use fractions or integers (but not decimals). (b) Apply the method of conjugate gradients to this problem. Use xo as the initial vector and obtain x. Here, you can use two decimal digits to write your final answers. #10(15 points). Do discrete least squares of the form + aCosx + aCos2x + bSinx for the data in [-,], shown below. Show your calculations clearly. x - - 2 y 10 5 02 -21 4 -1 -2] #11 (20 points). Matrix A = -1 4 -1. Initial vector xo=2 0 -1 4 (a) Perform two iterations of the Power Method, and estimate the largest eigenvalue? Use fractions or integers, but no decimals at all. (b) Explain how you will compute an eigenvalue closest to 2. (c) Draw Gerschgorin circles for A and estimate the spectral radius of A. [8] [8] 2 #12(15 points). Obtain a Householder matrix P such that Px=y, where x= and y= You must 0 show all the steps in constructing w and P. Use efficient calculations with integers or fractions only (no decimals at all). Finally write the 4x4 matrix P. Simplify the entries in P. (a) Use three-digit-chopping at each step, and Gaussian Elimination without pivoting, to solve the linear system, 0.011x+59.9y= 58.1; 5.44x-7.33y=48.1 (b) Use three-digit-rounding and Gaussian Elimination "with pivoting" to solve the same problem in part(a). Show the main steps in your Gaussian Elimination. #2(20 points). (a) If (I - A) is invertible and A=0, prove that (IA)=I+ A+ A. (b) Prove that the product of two orthogonal matrices A and B will also be orthogonal. (c) For the matrix A = [Cose Sine Sine], Compute A37, the thirty-seventh power of A, without -Cose actually multiplying out 37 times. Give reasons and show your steps. Do not use a calculator. (d) For some matrix C=[mm], find the largest singular value. Given m>0". [114] #3(15 points). The matrix A= 1 4 1 Obtain an LU Decomposition for this matrix. Do not use 411 decimals at all. Show your steps. Write the matrix L and U. #4(15 points). While solving the system, Ax=b, we came up with a formula for the relative error ||x-x|| as, ||x|| K(A), where x is an approximate solution, K(A) is the condition number and ||b||' r is the corresponding residual. (a) Prove this formula in your own words. Do not copy the proof from our book. (b) Explain the importance of this formula. #5(15 points). Fit a "least squares function P(x)" of the form P(x)= ACosx + Bx for the data shown below. If needed, round your final answers to 3 decimal digits. Show your normal equations and calculations clearly. x 0 2 y 1 3 10 #6(20 points). (a) Jacobi's method for solving a system Ax=b can be written in iterative form as: xk+1 = Txk+c. Suppose the method converges to the vector x, such that x=Tx+c. Prove that ||xn-x|| SITI (Recall that as n approaches infinity, Xn-x is the final error vector and xo-x is the ||Xox|| initial error vector). You can take any norm, L-2 or L-infinity. (b) Suppose the initial error vector is (xo-x)= [1 -2 2 0]T and the matrix T is symmetric with largest eigenvalue = 0.88. Use L norm to estimate the number of iterations required for Jacobi method to converge within 106 of the exact answer. #7(15 points). Find the "continuous" least squares polynomial of the form P(x) = A + Bx for the function f(x) = Cosx on the interval [0,]. Use two decimal digits in your final answer. For reference, the integral of xCosx is xSinx+Cosx. 3 4 2 #8(15 points). It is known that f(x) = x- x++... Compute a Pade' approximation for f(x), in the form: a+bx 1+cx+dx2' 4 -1 -1 0 4 0 #9(20 points). Suppose A = ,x= b= , xo-initial vector = 1 0 4 0 -1-1 4 1001 (a) Apply Gauss Seidel method to obtain vectors X1 and X2 in solving the system Ax=b. You must use fractions or integers (but not decimals). (b) Apply the method of conjugate gradients to this problem. Use xo as the initial vector and obtain x. Here, you can use two decimal digits to write your final answers. #10(15 points). Do discrete least squares of the form + aCosx + aCos2x + bSinx for the data in [-,], shown below. Show your calculations clearly. x - - 2 y 10 5 02 -21 4 -1 -2] #11 (20 points). Matrix A = -1 4 -1. Initial vector xo=2 0 -1 4 (a) Perform two iterations of the Power Method, and estimate the largest eigenvalue? Use fractions or integers, but no decimals at all. (b) Explain how you will compute an eigenvalue closest to 2. (c) Draw Gerschgorin circles for A and estimate the spectral radius of A. [8] [8] 2 #12(15 points). Obtain a Householder matrix P such that Px=y, where x= and y= You must 0 show all the steps in constructing w and P. Use efficient calculations with integers or fractions only (no decimals at all). Finally write the 4x4 matrix P. Simplify the entries in P.