12.1: Given/Find Blocks

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'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.

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?