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Chapter 2: Q2.3-42Q (page 93)

Exercises 42鈥�44 show how to use the condition number of a matrix Ato estimate the accuracy of a computed solution of \(Ax = b\). If the entries of Aand b are accurate to about rsignificant digits and if the condition number of Ais approximately \({\bf{1}}{{\bf{0}}^k}\) (with ka positive integer), then the computed solution of \(Ax = b\) should usually be accurate to at least \(r - k\) significant digits.

42. Find the condition number of the matrix A in Exercise 9. Construct a random vector x in \({\mathbb{R}^{\bf{4}}}\) and compute \({\bf{b}} = A{\bf{x}}\). Then use your matrix program to compute the solution \({{\bf{x}}_{\bf{1}}}\) of \(Ax = b\). To how many digits do x and \({{\bf{x}}_{\bf{1}}}\) agree? Find out the number of digits your matrix program stores accurately, and report how many digits of accuracy are lost when \({{\bf{x}}_{\bf{1}}}\) is

used in place of the exact solution x.

Short Answer

Expert verified

The solution has approximately 12 decimal places, and the calculated answer

(\({{\bf{x}}_1}\)) is accurate.

Step by step solution

01

Obtain the condition number of matrix A

Consider matrix A as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}4&0&{ - 3}&{ - 7}\\{ - 6}&9&9&9\\7&{ - 5}&{10}&{19}\\{ - 1}&2&4&{ - 1}\end{array}} \right]\)

Obtain thecondition numberof matrix A using the MATLAB command shown below:

\[\begin{array}{l} > > {\rm{ A }} = {\rm{ }}\left[ {{\rm{4 0 }} - {\rm{3 }} - {\rm{7; }} - {\rm{6 9 9 9; 7 }} - {\rm{5 10 19; }} - {\rm{1 2 4 }} - {\rm{1}}} \right];\\ > > {\rm{ C}} = {\rm{cond}}\left( {\rm{A}} \right)\end{array}\]

It gives the output 23683.

Thus, thecondition number of matrix A is 23683.

By comparing with thecondition number of A, that is \({10^k}\), the condition number is approximately \({10^4}\).

02

Obtain the solution by using the MATLAB command

It is found that x and\({{\bf{x}}_1}\)agree to at least 12 or 13significant digits if it run multiple experiments with MATLAB, which properly captures 16 digits.

Obtain a random matrix by using the MATLAB command shown below:

\( > > {\bf{x}} = {\rm{rand}}\left( {4,1} \right)\)

\({\bf{x}} = \left[ {\begin{array}{*{20}{c}}{0.9501}\\{0.2131}\\{0.6068}\\{0.4860}\end{array}} \right]\)

Now, compute\({\bf{b}} = A{\bf{x}}\)by using the MATLAB command shown below:

\(\begin{array}{l} > > {\rm{ }}A{\rm{ }} = {\rm{ }}\left[ {{\rm{4 0 }} - {\rm{3 }} - {\rm{7; }} - {\rm{6 9 9 9; 7 }} - {\rm{5 10 19; }} - {\rm{1 2 4 }} - {\rm{1}}} \right]{\rm{;}}\\ > > x = \left[ {0.9501{\rm{; 0}}{\rm{.2131; 0}}{\rm{.6068; 0}}{\rm{.4860}}} \right]{\rm{;}}\\ > > b = A*x\end{array}\)

The output is \({\bf{b}} = A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{ - 3.8493}\\{5.5795}\\{20.7973}\\{.8467}\end{array}} \right]\).

03

Obtain the MATLAB solution

Compute\({{\bf{x}}_1}\)of\(A{\bf{x}} = {\bf{b}}\)by using the MATLAB command shown below:

\(\begin{array}{l} > > {\rm{ }}A{\rm{ }} = {\rm{ }}\left[ {{\rm{4 0 }} - {\rm{3 }} - {\rm{7; }} - {\rm{6 9 9 9; 7 }} - {\rm{5 10 19; }} - {\rm{1 2 4 }} - {\rm{1}}} \right]{\rm{;}}\\ > > b = \left[ { - {\rm{3}}{\rm{.8493; 5}}{\rm{.5795; 20}}{\rm{.7973; 0}}{\rm{.8467}}} \right]{\rm{;}}\\ > > {x_1} = A\backslash b\end{array}\)

The output is\({{\bf{x}}_1} = \left[ {\begin{array}{*{20}{c}}{.9501}\\{.2311}\\{.6068}\\{.4860}\end{array}} \right]\).

Obtain the difference between x and\({{\bf{x}}_1}\).

\({\bf{x}} - {{\bf{x}}_1} = \left[ {\begin{array}{*{20}{c}}{.0171}\\{.4858}\\{ - .2360}\\{.2456}\end{array}} \right] \times {10^{ - 12}}\)

Thus, the solution has approximately 12 decimal places, and the calculated answer (\({{\bf{x}}_1}\)) is accurate.

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Most popular questions from this chapter

[M] For block operations, it may be necessary to access or enter submatrices of a large matrix. Describe the functions or commands of your matrix program that accomplish the following tasks. Suppose A is a \(20 \times 30\) matrix.

  1. Display the submatrix of Afrom rows 15 to 20 and columns 5 to 10.
  2. Insert a \(5 \times 10\) matrix B into A, beginning at row 10 and column 20.
  3. Create a \(50 \times 50\) matrix of the form \(B = \left[ {\begin{array}{*{20}{c}}A&0\\0&{{A^T}}\end{array}} \right]\).

[Note: It may not be necessary to specify the zero blocks in B.]

Describe in words what happens when you compute \({A^{\bf{5}}}\), \({A^{{\bf{10}}}}\), \({A^{{\bf{20}}}}\), and \({A^{{\bf{30}}}}\) for \(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

Suppose Tand U are linear transformations from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) such that \(T\left( {U{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\) . Is it true that \(U\left( {T{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\)? Why or why not?

Suppose the third column of Bis the sum of the first two columns. What can you say about the third column of AB? Why?

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5鈥�8, find formulas for X, Y, and Zin terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint:Compute the product on the left, and set it equal to the right side.]

8. \[\left[ {\begin{array}{*{20}{c}}A&B\\{\bf{0}}&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}X&Y&Z\\{\bf{0}}&{\bf{0}}&I\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&I\end{array}} \right]\]
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