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Chapter 6: Q-6-18SE (page 331)

In Exercises 17鈥�20, solve\(Ax = b\) and \(A\left( {\Delta x} \right) = \Delta b\) and show that the inequality (2) holds in each case. (See the discussion of ill-conditionedmatrices in Exercises 41鈥�43 in Section 2.3.)

\(18.\,\,A = \left[ {\begin{array}{*{20}{c}}{4.5}&{3.1}\\{1.6}&{1.1}\end{array}} \right],\,\,b = \left[ {\begin{array}{*{20}{c}}{.500}\\{ - 1.407}\end{array}} \right],\,\,\Delta b = \left[ {\begin{array}{*{20}{c}}{0.001}\\{ - 0.003}\end{array}} \right]\)

Short Answer

Expert verified

The condition \(\frac{{\parallel \Delta x\parallel }}{{\parallel x\parallel }} \le cond(A) \cdot \frac{{\parallel \Delta b\parallel }}{{\parallel b\parallel }}\)is satisfied.

Step by step solution

01

The condition for solving matrix

Enter the matrices in MATLAB

\(\begin{array}{c}A = \left[ {4.5{\rm{ }}3.1{\rm{ }};{\rm{ }}1.6{\rm{ }}1.1} \right]\\b = \left[ {0.500;{\rm{ - 1}}{\rm{.407}}} \right]\\delta\,\,b = \left[ {0.001; - 0.003} \right]\end{array}\)

02

Solve the equation matrix

Solve the system of equation

\(\)

\(\begin{array}{c}Ax = b\\x = {A^{ - 1}}b\\x{\rm{ }} = \left[ { - 491.17,\,\,731.15} \right]\end{array}\)

Solve the system of equation as:

\(\begin{array}{c}A\left( {delta\,\,x} \right) = delta\,\,b\\delta\,\,x = {A^{ - 1}}\left( {delta\,\,b} \right)\\delta\,\,x = \left[ { - 1.04,\,\,1.51} \right]\end{array}\)

03

Find norm and condition number 

\(\begin{array}{l}norm\left( {delta\,\,x} \right)/norm\left( x \right)\\ans{\rm{ }} = 0.0021\\norm\left( {delta\,\,b} \right)/norm(b)\\ans{\rm{ }} = 0.0021\end{array}\)

\(\begin{array}{l}ans{\rm{ }} = \\1.5479e - 04\\Condition{\rm{ }}Number{\rm{ }}of{\rm{ }}A:\\cond\left( A \right)\\\end{array}\)

Condition Number of A:

\(\begin{array}{l}Cond\left( A \right):\\ans{\rm{ }} = 3.3630e + 03\end{array}\)

04

Find ratio condition 

Compute the ratio condition as:

\(\begin{array}{l}(A) \cdot \frac{{\parallel \Delta b\parallel }}{{\parallel b\parallel }}\\cond\left( A \right)*{\rm{ }}norm\left( {deltab} \right)/norm\left( b \right)\\ans{\rm{ }} = 7.1221\end{array}\)

So, the condition\(\frac{{\parallel \Delta x\parallel }}{{\parallel x\parallel }} \le cond(A) \cdot \frac{{\parallel \Delta b\parallel }}{{\parallel b\parallel }}\)satisfied.

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Most popular questions from this chapter

In Exercises 17 and 18, all vectors and subspaces are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

17. a.If \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) is an orthogonal basis for\(W\), then multiplying

\({v_3}\)by a scalar \(c\) gives a new orthogonal basis \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},c{{\bf{v}}_3}} \right\}\).

b. The Gram鈥揝chmidt process produces from a linearly independent

set \(\left\{ {{{\bf{x}}_1}, \ldots ,{{\bf{x}}_p}} \right\}\)an orthogonal set \(\left\{ {{{\bf{v}}_1}, \ldots ,{{\bf{v}}_p}} \right\}\) with the property that for each \(k\), the vectors \({{\bf{v}}_1}, \ldots ,{{\bf{v}}_k}\) span the same subspace as that spanned by \({{\bf{x}}_1}, \ldots ,{{\bf{x}}_k}\).

c. If \(A = QR\), where \(Q\) has orthonormal columns, then \(R = {Q^T}A\).

Let \({\mathbb{R}^{\bf{2}}}\) have the inner product of Example 1, and let \({\bf{x}} = \left( {{\bf{1}},{\bf{1}}} \right)\) and \({\bf{y}} = \left( {{\bf{5}}, - {\bf{1}}} \right)\).

a. Find\(\left\| {\bf{x}} \right\|\),\(\left\| {\bf{y}} \right\|\), and\({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).

b. Describe all vectors\(\left( {{z_{\bf{1}}},{z_{\bf{2}}}} \right)\), that are orthogonal to y.

In exercises 1-6, determine which sets of vectors are orthogonal.

  1. \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\4\\{ - 3}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}5\\2\\1\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}3\\{ - 4}\\{ - 7}\end{array}} \right]\)

In Exercises 1-6, the given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

5. \(\left( {\begin{aligned}{{}{}}1\\{ - 4}\\0\\1\end{aligned}} \right),\left( {\begin{aligned}{{}{}}7\\{ - 7}\\{ - 4}\\1\end{aligned}} \right)\)

Let \({\mathbb{R}^{\bf{2}}}\) have the inner product of Example 1. Show that the Cauchy-Schwarz inequality holds for \({\bf{x}} = \left( {{\bf{3}}, - {\bf{2}}} \right)\) and \({\bf{y}} = \left( { - {\bf{2}},{\bf{1}}} \right)\). (Suggestion: Study \({\left| {\left\langle {{\bf{x}},{\bf{y}}} \right\rangle } \right|^{\bf{2}}}\).)
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