Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Textbook Question
Chapter 2.5, Problem 3CP
Rewrite Program 2.2 to carry out Gauss-Seidel iteration, Solve the problem in Example 2.24 to check your work.
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3. Bayesian Inference – Updating Beliefs
A medical test for a rare disease has the following characteristics:
Sensitivity (true positive rate): 99%
Specificity (true negative rate): 98%
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a) Define the relevant events and use Bayes’ Theorem to compute the probability that the patient actually has the disease.b) Explain why the result might seem counterintuitive, despite the high sensitivity and specificity.c) Discuss how prior probabilities influence posterior beliefs in Bayesian inference.d) Suppose a second, independent test with the same accuracy is conducted and is also positive. Update the probability that the patient has the disease.
answer number 6
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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