Question

[M] Use equation (6) to solve the problem in Exercise 13. Set \({{\bf{x}}^{\left( {\bf{0}} \right)}} = {\bf{d}}\), and for \[k = {\bf{1}},\,{\bf{2}},\,....\] compute \({{\bf{x}}^{\left( k \right)}} = {\bf{d}} + C{{\bf{x}}^{\left( {k - {\bf{1}}} \right)}}\). How many steps are needed to obtain the answer in Exercise 13 to four significant figures?

Short answer

12 steps

Step-by-step solution

Write the augmented matrix

The augmented matrix is shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{.8412}&{ - 0.0064}&{ - 0.0025}&{ - 0.0304}&{ - 0.0014}&{ - 0.0083}&{ - 0.1594}\\{ - .0057}&{0.7355}&{ - 0.0436}&{ - 0.0099}&{ - 0.0083}&{ - 0.0201}&{ - 0.3413}\\{ - .0264}&{ - 0.1506}&{0.6443}&{ - 0.0139}&{ - 0.0142}&{ - 0.0070}&{ - 0.0236}\\{ - .3299}&{ - 0.0565}&{ - 0.0495}&{0.6364}&{ - 0.0204}&{ - 0.0483}&{ - 0.0649}\\{ - .0089}&{ - 0.0081}&{ - 0.0333}&{ - 0.0295}&{0.6588}&{ - 0.0237}&{ - 0.0020}\\{ - 0.1190}&{ - 0.0901}&{ - 0.0996}&{ - 0.1260}&{ - 0.1722}&{0.7632}&{ - 0.3369}\\{ - 0.0063}&{ - 0.0126}&{ - 0.0196}&{ - 0.0098}&{ - 0.0064}&{ - 0.0132}&{0.9988}\end{array}}&{\begin{array}{*{20}{c}}{74000}\\{56000}\\{10500}\\{25000}\\{17500}\\{196000}\\{5000}\end{array}}\end{array}\,} \right]\)

Write the iterations

\({{\bf{x}}^{\left( 0 \right)}} = \left( {74000.0,\,56000.0,\,10500.0,\,25000.0,\,17500.0,\,196000.0,\,5000.0} \right)\)

\({{\bf{x}}^{\left( 1 \right)}} = \left( {89344.2,\,77730.5,\,26708.1,\,72334.7,\,30325.6,\,265158.2,\,9327.8} \right)\)

\({{\bf{x}}^{\left( 2 \right)}} = \left( {94681.2,\;87714.5,\;37577.3,\;100520.5,\;38598.0,\;296563.8,\;11480.0} \right)\)

\({{\bf{x}}^{\left( 3 \right)}} = \left( {97091.9,\;92573.1,\;43867.8,\;115457.0,\;43491.0,\;312319.0,\;12598.8} \right)\)

\({{\bf{x}}^{\left( 4 \right)}} = \left( {98291.6,\;95033.2,\;47314.5,\;123202.5,\;46247.0,\;320502.4,\;13185.5} \right)\)

\({{\bf{x}}^{\left( 5 \right)}} = \left( {98907.2,\;96305.3,\;49160.6,\;127213.7,\;47756.4,\;324796.1,\;13655.9} \right)\)

\({{\bf{x}}^{\left( 6 \right)}} = \left( {99226.6,\;96969.6,\;50139.6,\;129296.7,\;48569.3,\;327053.8,\;13655.9} \right)\)

\({{\bf{x}}^{\left( 7 \right)}} = \left( {99393.1,\;97317.8,\;50928.7,\;130948.0,\;49002.8,\;328240.9,\;13741.1} \right)\)

\({{\bf{x}}^{\left( 8 \right)}} = \left( {99480.0,\;97500.7,\;50928.7,\;130948.0,\;49232.5,\;328864.7,\;13785.9} \right)\)

\({{\bf{x}}^{\left( 9 \right)}} = \left( {99525.5,\;97647.2,\;51147.2,\;131399.2,\;49417.7,\;329364.4,\;13821.7} \right)\)

\({{\bf{x}}^{\left( {10} \right)}} = \left( {99561.9,\;97673.7,\;51186.8,\;131480.4,\;49451.3,\;329454.7,\;13828.2} \right)\)

\({{\bf{x}}^{\left( {11} \right)}} = \left( {99561.9,\;97687.6,\;51186.8,\;131480.4,\;49451.3,\;329502.1,\;13831.6} \right)\)

\({{\bf{x}}^{\left( {12} \right)}} = \left( {99568.4,\;97687.6,\;51207.5,\;131523.0,\;49469.0,\;329502.1,\;13831.6} \right)\)

So, the answer to Exercise 13 is obtained in 12 steps.