12.1: Given/Find Blocks
- Page ID
- 14985
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)'given/find' block
We illustrate this through the following example.
Example 12.1.1
Solve: 2 = x4 + 3x2 − 1.
Solution
In the worksheet, the syntax is
given
2 = x4+3x2- 1 (Use ctrl + = for bold symbolic equals sign)
guess
x := 1
x := find(x)
x =
The last line will show x = 0.89 once the equals sign is entered. The line x := find(x) solves for x and then stores it back as the variable x. For full details, the guess of x := 1 provides the “seed” for the Newton-Raphson algorithm (Google it). Try changing the seed value to see what happens. For example, if we start with x := −1 we end up with a different answer, namely x = −0.89.
The given/find process can be extended to solving systems of equations as well as in the next example:
Example 12.1.2
Simultaneously solve a + b = 3, a − b = 4 for a, b.
Solution
In the worksheet, using ctrl + = for bold symbolic equals sign, the syntax is
given
a+b 3
a-b 4
guess
a := 1
b := 1
find(a,b)=
Once, you hit return on the last line, it will become
\[find(a,b)= \begin{pmatrix} 3.5 \\ -0.5 \end{pmatrix}\nonumber\]
The given/find capabilities are not limited to linear equations as can be seen in the next example.
The given/find capabilities are not limited to linear equations as can be seen in the next example.
Example 12.1.3
Simultaneously solve t4+r3 = 1 and r−t2 = −2
Solution
In the worksheet, using ctrl + = for bold symbolic equals sign, the syntax is
given
t4+r3 = 1
r−t2 = −2
guess
r := 1
t := 1
find(r,t)=
Once, you hit return on the last line, it will become
\[find(r,t)= \begin{pmatrix} -0.783 \\ 1.103 \end{pmatrix}\nonumber\]
So, one solution is r = −0.783, t = 1.103. Note that this is not the only solution. Can you change the seed (or “guess”) values to get the rest of the solutions?