10.9 Solve the following set of equations with LU decomposition: 3×1 2×2 x3 10
2×1 6×2 4×3 44 x1 2×2 5×3 26
10.10 a Determine the LU decomposition without pivoting by hand for the following matrix, and check your results by validating that LU A.
How many digits of precision will be lost due to illconditioning? b Repeat part a, but scale the matrix by making the maximum element in each row equal to 1.
10.16 Determine the condition number based on the rowsum norm for the normalized 4 4 Hilbert matrix. How many signifi cant digits of precision will be lost due to illconditioning?
10.17 Besides the Hilbert matrix, there are other matrices that are inherently illconditioned. One such case is the Vandermonde matrix, which has the following form:
6 2 A 8 1.1
Scale the matrix by making the maximum element in each row equal to 1.
10.13 Determine the Frobenius and the rowsum norms for the systems in Probs. 10.3 and 10.4. Scale the matrices by making the maximum element in each row equal to 1.
10.21 Consider these vectors:
10.14 A matrix A is defined as
0.125 0.25 0.5 1
A2i 3j ak B bi j 4k C3i cj 2k
Using the columnsum norm, compute the condition number and the number of suspect digits generated by this matrix.
10.15 aDeterminetheconditionnumberforthefollowingsystem using the rowsum norm. Do not normalize the system.
Vector A is perpendicular to B as well as to C. It is also known that
0.015625 0.625 0.25 1
B C 2. Use any method presented in this chapter to solve for the three unknowns, a, b, and c.
10.22 Consider the following vectors:
149 4916 9 16 25 16 25 36 25 36 49
16 25 25 36 36 49 49 64 64 81
A ai bj ck B 2i j 4k C i 3j 2k
where A is an unknown vector. If
A B A C 5 a 6 i 3 b 2 j 4 c 1 k
use any method covered in this chapter to solve for the three un knowns, a, b, and c.
0.00463 0.02777 0.16667 1
0.001953 0.015625 0.125 1
x21 x1 1 3 7 2 x2 x2 1
b Employ the result of a to compute the determinant.
c Repeat a and b using MATLAB.
10.11 Use the following LU decomposition to a compute the de terminant and b solve Ax b with bT 10 44 26.
1 3 2 1 0.6667 1 7.3333 4.6667
a Determine the condition number based on the rowsum norm for the case where x1 4, x2 2, and x3 7.
b Use MATLAB or Mathcad software to compute the spectral and Frobenius condition numbers.
10.18 Develop a userfriendly program for LU decomposition based on the pseudocode from Fig. 10.2.
10.19 Develop a userfriendly program for LU decomposition, in cluding the capability to evaluate the matrix inverse. Base the pro gram on Figs. 10.2 and 10.5.
10.20 Use iterative refinement techniques to improve x1 2, x2 3, and x3 8, which are approximate solutions of
2×1 5×2 x3 5 5×1 2×2 x3 12 x1 2×2 x3 3
10.12 Determine Ae, A1, and A for
0.3333 0.3636 1
304 LU DECOMPOSITION AND MATRIX INVERSION
10.23 Let a function be defined on the interval 0, 2 as follows:
Use sufficient precision in displaying results to allow you to
detect imprecision.
b Repeat part a using a 7 7 Hilbert matrix.
c Repeat part a using a 10 10 Hilbert matrix.
10.25 Polynomial interpolation consists of determining the unique n 1thorder polynomial that fits n data points. Such polynomi als have the general form,
Determine the constants a, b, c, and d so that the function f satisfies
cx d, 1 x 2 the following:
f0 f2 1.
f is continuous on the entire interval. a b 4.
Derive and solve a system of linear algebraic equations with a ma trix form identical to Eq. 10.1.
a Create a 3 3 Hilbert matrix. This will be matrix A. Multiply
the matrix by the column vector x 1, 1, 1T. The solution of Ax will be another column vector b. Using any numerical package and Gauss elimination, find the solution to Ax b using the Hilbert matrix and the vector b that you calculated. Compare the result to the known x vector.
fx p1xn1 p2xn2 … pn1 x pn
where the ps are constant coefficients. A straightforward way for computing the coefficients is to generate n linear algebraic equa tions that we can solve simultaneously for the coefficients. Suppose that we want to determine the coefficients of the fourthorder poly nomial fx p1x4 p2x3 p3x2 p4x p5 that passes through the following five points: 200, 0.746, 250, 0.675, 300, 0.616, 400, 0.525, and 500, 0.457. Each of these pairs can be substituted into Eq. P10.25 to yield a system of five equations with five un knowns the ps. Use this approach to solve for the coefficients. In addition, determine and interpret the condition number.