Science:Math Exam Resources/Courses/MATH307/April 2009/Question 01 (d)

MATH307 April 2009

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[hide] Question 01 (d)
In trying to solve the equation Ax = b, where A is the matrix in part (c), $A={\begin{bmatrix}-10&0&0\\0&-3.5&0\\0&0&0.5\end{bmatrix}}$ you are able to measure b with an accuracy of ${\frac {\|\Delta b\|}{\|b\|}}\leq 0.1$. What is the largest possible relative error ${\frac {\|\Delta x\|}{\|x\|}}$ in your solution for x?

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[show] Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. METHODOLOGY Suppose the measured data is given by: ${\vec {b'}}={\vec {b}}+{\vec {\Delta b}}$ where ${\vec {b}}$ is the true data, ${\vec {\Delta b}}$ is the error in the data. If a vector ${\vec {x}}$ was the true solution then $A{\vec {x}}={\vec {b}}$. Using ${\vec {b'}}$ we can get an approximate solution for ${\vec {x}}$ given by: ${\vec {x'}}=A^{-1}{\vec {b'}}$ Substituting in ${\vec {b'}}={\vec {b}}+{\vec {\Delta b}}$, we get: ${\vec {x'}}=A^{-1}({\vec {b}}+{\vec {\Delta b}})$ ${\vec {x'}}=A^{-1}{\vec {b}}+A^{-1}{\vec {\Delta b}}$ ${\vec {x'}}=A^{-1}{\vec {b}}+A^{-1}{\vec {\Delta b}}={\vec {x}}+{\vec {\Delta x}}$ This gives us: $A^{-1}{\vec {\Delta b}}={\vec {\Delta x}}$ which we can use to find the upper bound of the relative error ${\frac {\|\Delta x\|}{\|x\|}}$. To do this we will start with: $\|{\vec {\Delta x}}\|=\|A^{-1}{\vec {\Delta b}}\|$ ...and then multiply both sides by $\|{\vec {b}}\|$ to get: $\|{\vec {\Delta x}}\|\|{\vec {b}}\|=\|A^{-1}{\vec {\Delta b}}\|\|{\vec {b}}\|$ We know that $\|{\vec {b}}\|=\|A{\vec {x}}\|$ and we can substitute this into the above equation to get: $\|{\vec {\Delta x}}\|\|{\vec {b}}\|=\|A^{-1}{\vec {\Delta b}}\|\|{\vec {b}}\|=\|A^{-1}{\vec {\Delta b}}\|\|A{\vec {x}}\|$ It can be proven that that for any matrix A and any vector x, the equality $\|A{\vec {x}}\|\leq \|A\|\|{\vec {x}}\|$ will always hold true. We can use this in solving for the relative error. Using this equality, we get: $\|{\vec {\Delta x}}\|\|{\vec {b}}\|\leq \|A^{-1}{\vec {\Delta b}}\|\|A{\vec {x}}\|$ Divide both sides by $\|{\vec {x}}\|\|{\vec {b}}\|$: ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}\leq \|A\|\|A^{-1}\|{\frac {\|\Delta {\vec {b}}\|}{\|{\vec {b}}\|}}$ Since $\|A\|\|A^{-1}\|=cond(A)$: ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}\leq cond(A){\frac {\|\Delta {\vec {b}}\|}{\|{\vec {b}}\|}}$ We have finished finding an equation which tells us an upper bound on the relative error ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}$. This says the larger the condition number, the less control we have on the relative error. ANSWER To solve this question we have to substitute in ${\frac {\|\Delta b\|}{\|b\|}}$ and $cond(A)$. The question gives ${\frac {\|\Delta b\|}{\|b\|}}\leq 0.1$, but we want the largest possible relative error ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}$. Since ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}$ is proportional to ${\frac {\|\Delta b\|}{\|b\|}}$, we want the largest value of ${\frac {\|\Delta b\|}{\|b\|}}$ which is 0.1, so we will use ${\frac {\|\Delta b\|}{\|b\|}}=0.1$. From the part (c) of question 1, we got that $cond(A)=20$, and we have both values needed to solve for the relative error. ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}\leq cond(A){\frac {\|\Delta {\vec {b}}\|}{\|{\vec {b}}\|}}$ ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}\leq 20\cdot 0.1$ ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}\leq 2$ FINAL ANSWER ${\frac {\|\Delta {\vec {x}}\|}{\|{\vec {x}}\|}}\leq 2$