Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Textbook Question
Chapter 2.5, Problem 5CP
Carry out the steps of Computer Problem 1 with
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(12.1)
SOLVE
30 KULTURT
2
x²y" + xy² + (1x² - 2) y = 0 BY BECIELS METHOD.
#1 Solve the following non-linear equations manually using:
a) Bisection Method (5 iterations)
Answer by (True) or (False) Only:
1. The next iterative value of the root of (X²– 4 = 0) using the Newton-Raphson method, if the initial guess
is (3), is (2.166).
2. In the Gauss elimination method, the given system is transformed into an equivalent system with lower
- triangular matrix.
3. The 1s' positive root of equation (tanx – 2tanhx= 0) occurs in the interval [0,1].
4. The process of finding the value of (x), corresponding to a given (y) which is not in the table, is called
" Inverse Interpolation ".
5. In bisection method if root lies between (a) and (b), then f (a) × f (b) is < 0.
Chapter 2 Solutions
Numerical Analysis
Ch. 2.1 - Use Gaussian elimination to solve the systems:...Ch. 2.1 - Use Gaussian elimination to solve the systems:...Ch. 2.1 - Solve by back substitution: a.3x4y+5z=23y4z=15z=5...Ch. 2.1 - Solve the tableau form a.[ 34236612382-1 ] b.[...Ch. 2.1 - Use the approximate operation count 2n3/3 for...Ch. 2.1 - Assume that your computer completes a 5000...Ch. 2.1 - Assume that a given computer requires 0.002...Ch. 2.1 - If a system of 3000 equations in 3000 unknowns can...Ch. 2.1 - Put together the code fragments in this section to...Ch. 2.1 - Let H denote the nn Hubert matrix, whose (i,j)...
Ch. 2.2 - Find the LU factorization of the given matrices....Ch. 2.2 - Find the LU factorization of the given matrices....Ch. 2.2 - Solve the system by finding the LU factorization...Ch. 2.2 - Solve the system by finding the LU factorization...Ch. 2.2 - Solve the equation Ax=b, where A=[...Ch. 2.2 - Given the 10001000 matrix A, your computer can...Ch. 2.2 - Assume that your computer can solve 1000 problems...Ch. 2.2 - Assume that your computer can solve a 20002000...Ch. 2.2 - Let A be an nn matrix. Assume that your computer...Ch. 2.2 - Use the code fragments for Gaussian elimination in...Ch. 2.2 - Add two-step back substitution to your script from...Ch. 2.3 - Find the norm A of each of the following...Ch. 2.3 - Find the (infinity norm) condition number of (a)...Ch. 2.3 - Find the forward and backward errors, and the...Ch. 2.3 - Find the forward and backward errors and error...Ch. 2.3 - Find the relative forward and backward errors and...Ch. 2.3 - Find the relative forward and backward errors and...Ch. 2.3 - Find the norm H of the 55 Hilbert matrix.Ch. 2.3 - (a) Find the condition number of the coefficient...Ch. 2.3 - (a) Find the condition number (in the infinity...Ch. 2.3 - (a) Find the (infinity norm) condition number of...Ch. 2.3 - (a) Prove that the infinity norm x is a vector...Ch. 2.3 - (a) Prove that the infinity norm A is a matrix...Ch. 2.3 - Prove that the matrix infinity norm is the...Ch. 2.3 - Prove that the matrix 1-norm is the operator norm...Ch. 2.3 - For the matrices in Exercise 1, find a vector x...Ch. 2.3 - For the matrices in Exercise 1, find a vector...Ch. 2.3 - Prob. 17ECh. 2.3 - Prob. 18ECh. 2.3 - For the nn matrix with entries Aij=5/(i+2j1), set...Ch. 2.3 - Carry out Computer Problem 1 for the matrix with...Ch. 2.3 - Let A be the nn matrix with entries Aij=| ij |+1 ....Ch. 2.3 - Carry out the steps of Computer Problem 3 for the...Ch. 2.3 - For what values of n does the solution in Computer...Ch. 2.3 - Use the MATLAB program from Computer Problem 2.1.1...Ch. 2.4 - Find the PA=LU factorization (using partial...Ch. 2.4 - Find the PA=LU factorization (using partial...Ch. 2.4 - Solve the system by finding the PA=LU...Ch. 2.4 - Solve the system by finding the PA=LU...Ch. 2.4 - Write down a 55 matrix P such that multiplication...Ch. 2.4 - (a) Write down the 44 matrix P such that...Ch. 2.4 - Change four entries of the leftmost matrix to make...Ch. 2.4 - Find the PA=LU factorization of the matrix A in...Ch. 2.4 - (a) Find the PA=LU factorization of A=[...Ch. 2.4 - (a) Assume that A is an nn matrix with entries |...Ch. 2.4 - Write a MATLAB program to define the structure...Ch. 2.4 - Plot the solution from Step 1 against the correct...Ch. 2.4 - Rerun the calculation in Step 1 for n=102k, where...Ch. 2.4 - Add a sinusoidal pile to the beam. This means...Ch. 2.4 - Rerun the calculation as in Step 3 for the...Ch. 2.4 - Now remove the sinusoidal load and add a 70 kg...Ch. 2.4 - If we also fix the free end of the diving board,...Ch. 2.4 - Ideas for further exploration: If the width of the...Ch. 2.5 - Compute the first two steps of the Jacobi and the...Ch. 2.5 - Rearrange the equations to form a strictly...Ch. 2.5 - Apply two steps of SOR to the systems in Exercise...Ch. 2.5 - Apply two steps of SOR to the systems in Exercise...Ch. 2.5 - Let be an eigenvalue of an nn matrix A. (a) Prove...Ch. 2.5 - Use the Jacobi Method to solve the sparse system...Ch. 2.5 - Use the Jacobi Method to solve the sparse system...Ch. 2.5 - Rewrite Program 2.2 to carry out Gauss-Seidel...Ch. 2.5 - Rewrite Program 2.2 to carry out SOR. Use =1.1 to...Ch. 2.5 - Carry out the steps of Computer Problem 1 with...Ch. 2.5 - Prob. 6CPCh. 2.5 - Using your program from Computer Problem 3. decide...Ch. 2.6 - Show that the following matrices are symmetric...Ch. 2.6 - Show that the following symmetric matrices are not...Ch. 2.6 - Prob. 3ECh. 2.6 - Show that the Cholesky factorization procedure...Ch. 2.6 - Prob. 5ECh. 2.6 - Find the Cholesky factorization A=RTR of each...Ch. 2.6 - Prob. 7ECh. 2.6 - Solve the system of equations by finding the...Ch. 2.6 - Prob. 9ECh. 2.6 - Find all numbers d such that A=[ 122d ] is...Ch. 2.6 - Prob. 11ECh. 2.6 - Prove that a principal submatrix of a symmetric...Ch. 2.6 - Solve the problems by carrying out the Conjugate...Ch. 2.6 - Solve the problems by carrying out the Conjugate...Ch. 2.6 - Carry out the conjugate gradient iteration in the...Ch. 2.6 - Prob. 1CPCh. 2.6 - Use a MATLAB version of conjugate gradient to...Ch. 2.6 - Solve the system Hx=b by the Conjugate Gradient...Ch. 2.6 - Solve the sparse problem of (2.45) by the...Ch. 2.6 - Prob. 5CPCh. 2.6 - Let A be the nn matrix with n=1000 and entries...Ch. 2.6 - Prob. 7CPCh. 2.6 - Prob. 8CPCh. 2.6 - Prob. 9CPCh. 2.6 - Prob. 10CPCh. 2.7 - Find the jacobian of the functions a....Ch. 2.7 - Use the Taylor expansion to find the linear...Ch. 2.7 - Sketch the two curves in the uv-plane, and find...Ch. 2.7 - Apply two steps of Newtons Method to the systems...Ch. 2.7 - Apply two steps of Broyden I to the systems in...Ch. 2.7 - Prob. 6ECh. 2.7 - Prove that (2.55) satisfies (2.53) and (2.54).Ch. 2.7 - Prove that (2.58) satisfies (2.56) and (2.57).Ch. 2.7 - Implement Newtons Method with appropriate starting...Ch. 2.7 - Use Newtons Method to find the three solutions of...Ch. 2.7 - Use Newtons Method to find the two solutions of...Ch. 2.7 - Apply Newtons Method to find both solutions of the...Ch. 2.7 - Use Multivariate Newtons Method to find the two...Ch. 2.7 - Prob. 6CPCh. 2.7 - Apply Broyden I with starting guesses x0=(1,1) and...Ch. 2.7 - Apply Broyden II with starting guesses (1, 1) and...Ch. 2.7 - Prob. 9CPCh. 2.7 - Apply Broyden Ito find the intersection point in...Ch. 2.7 - Apply Broyden II to find the sets of two...Ch. 2.7 - Apply Broyden II to find the intersection point in...
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- mm.2arrow_forwardQ.2 Answer by (True) or (False) Only: 1. The next iterative value of the root of (X2-4 = 0) using the Newton-Raphson method, if the initial is (3), is (2.166). 2. In the Gauss elimination method, the given system is transformed into an equivalent system with lower - triangular matrix. 3. The 1st positive root of equation (tanx - 2tanhx = 0) occurs in the interval [0,1]. 4. The process of finding the value of (x), corresponding to a given (y) which is not in the table, is called " Inverse Interpolation ". 5. In bisection method if root lies between (a) and (b), then f (a) × f (b) is < 0. guessarrow_forward3. Using Gauss Jacobi method, solve the real roots of the given equation/s • Perform iteration until error is <1% • 5w – x + y + 3z = 11.9 • - 2w + 3x - y + 2z = -2 • 3w + x - 3y + 5z = 14.1 • 2w – 4x + y + 2z = 4.4arrow_forward
- 2. Use the Newton-Raphson method to solve, with initial guess x₁ = x₂ = 1 f₁(x)=x²-x₂-1=0 f(x)=x²-x₁-1=0 Do two iterations only.arrow_forwardConsider the system [BAH[3] = 21 (a) Use the starting guess X(0) 1.5 show that X(¹) -([4] = Find X (2) and X (3) (b) Use the starting guess X(0) to show that X(¹) = = 1.5 -3.25 A in an implementation of the Jacobi method to in an implementation of the Gauss-Seidel method Find X(²) and X (3)arrow_forwardGiven the following system and using the Jacobi method, answer the following questions = 10 (1) T1 + 5x2 + 9x3 Зд1 + 622 + 23 — 3 (2) 8х1 + 72 — 5аз — 1 (3) A. Given the initial values (x, x,", x) = (1,1,1), then what is the value of r after first iteration (0) (1) B. Using the data from part (A), what is the value of x (2) C. Using the data from part (A), what is the value of a after 2nd iteration (2) D. Using the data from part (A), what is the value of x after 2nd iteration E. What is %REaproz for the variable a, between the first and second iterations. Note; if your answer is 80.91%, then write 80.91arrow_forward
- 3.3 5x - 2y + z = 4 7x + y - 5z = 8 3x + 7y + 4z = 10) x + 5y + z = 14 3.4 2xy + 8z = 12 4x + y + 3z = 17 4aß +y = 8) 4α by using LU-Factorization Method. by Gauss Jacobi Iterative Method up to four iterations (Starting with x(0) = 1, y = 1, z0) = 1) 3.5 2a +58 +2y = 3 by Gauss-Seidel iterative technique up to three iterations a + 2B + 4y = 11) 4. Answer the following Questions and Calculate Relative Error and Percentage Error in each case: 4.1 Use the Bisection Method to find a solution of the equation 3x - e* = 0 on the interval [1,2] with an accuracy of 10-³. 4.2 Use the Newton-Raphson Method to find a solution accurate to within 10-6 for the following equation: sin x = ex, for 0≤x≤1 4.3 Use the Second Method to find an approximation to (29)1/4, accurate to within 10-4. 5. Use Newton-Iterative Techniques to find approximate solutions up to 2nd iterations to the given nonlinear system of equations. Choose: (xo, Yo) = (3,2) x² - y² = 4 x² + y² = 16arrow_forwardSolve no. 15 using matlab codesarrow_forward#1 Solve the following non-linear equations manually using: b) Regula Falsi / False-Position Method (5 iterations)arrow_forward
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