61cfcb96-3dca-427c-a2a5-6a0792ec6552 17 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Using graphs in solving equations:

]]>

In this example we seek the solution(s) for the equation:

]]>

x^2 + 1 = x^3 +2 *x -5

]]>

A solution can be found by determining the intersection of the two functions:

x f x 2^1 + :

x g x 3^2 x * + 5 - :

Using graphics and zooming the intersection we estimate the solution to be close to x = 1.80

x f x g 2 1 sys x f x g 2 1 sys

Use plot scale and move

The exact solution can be found using: note use Solve 2 and Boolean = sign

x 2^1 + x 3^2 x * + 5 - ≡ x solve #

Using Matrix Math to Solve Equations

]]>

A system of linear equations may be represented in matrix form. Thus the system of two equations

x – 2y = –1

3x + 4y = 17

may be represent by

]]>

12 - 3422 mat x y 21 mat * 1 - 1721 mat ≡

Let us write this in the form: AX = C

Multiply both sides by A^-1 giving A^-1AX =A^-1C

But since A^-1A = I, this becomes IX = A^-1C (I being the identify matrix)

We know that IX = X, therefore X = A^-1C

From this we see that the value of the X matrix may be obtained by computing A^–1C. The following figure

shows a SMath implementation of this method.

]]>

x 2 y * - 1 - ≡

3 x * 4 y * + 17 ≡

A 12 - 3422 mat :

C 1 - 1721 mat :

A.inver A invert :

A.inver 0.4 0.2 0.3 - 0.1 22 mat

X A.inver C * :

X 3221 mat

so solution is x=3 and y=2

to check

A X * 1 - 1721 mat

Another example

]]>

2x + 3y - 2z = 15 3x - 2y + 2z = -2 4x - y + 3z = 2

AA 232 - 32 - 241 - 333 mat :

CC 152 - 231 mat :

AA invert CC * 231 - 31 mat

x = 2 y = 3 z = -1

Function defined by calculus operations

]]>

x t h x 2^t t ln * x / + t diff :

x t h 1 t ln + x /

23 h 1.0493

x r 1 t + (2^t 0 x int :

2 r 8.6667

1 x + ( 2 ^

Solving single equations with solve

Function solve can be used to solve single equations.

The function can be called using

either of the following calls:

solve(equation, variable)

solve(equation, variable, lower limit, upper limit)

]]>

Examples using both symbolic and numeric results are shown below:

]]>

x 3^18 - 0 ≡ x solve #

Note: to enter the Boolean equal sign (bold =) use

Ctrl+=, or the symbol in the Boolean palette.

]]>

x 2^18 - 0 ≡ x 5 - 5 solve 424264087468461100000000000000 / - 5303300818672712500000000000 / 21 mat

x 2^18 - 0 ≡ x 5 - 5 solve 4.2426 - 4.2426 21 mat

x 3^18 - 0 ≡ x 5 - 5 solve 2.6207

x cubed gives only one real root

Solving polynomial equations with polyroots

Function polyroots returns the roots of a polynomial using as argument a column vector

with the polynomial coefficients. The coefficients must be entered in the vector in

increasing order of the power of the independent variable. For example, the vector

corresponding to the polynomial is:

]]>

18 - 00141 mat polyroots 2.6207 1.3104 - 2.2696 i * + 1.3104 - 2.2696 i * - 31 mat

12 x * + 5 x 2^* -

v 125 - 31 mat :

v polyroots 0.2899 - 0.6899 21 mat

x 0.6899 :

12 x * + 5 x 2^* - 1.005 105 -^* -

which is zero

Unlike function solve, which provides only real functions, function polyroots provides

either real or complex roots, or both. For example:

]]>

23 x * - 5 x 2^* + 7 x 3^* +

w 23 - 5741 mat :

w polyroots 0.2647 0.3996 i * + 0.2647 0.3996 i * - 1.2436 - 31 mat

let

x 1.2436 - :

23 x * - 5 x 2^* + 7 x 3^* + 0.0006

When we speak of solving an equation we mean: find the value(s) of x that satisfy the equation f(x) = g(x).

For example we might wish to know what value of x gives the function exp(x²+x) the value of 10.

When we speak of finding the roots of f(x) we mean finding what values of x makes f(x) = 0.

We will start this topic showing the solution to a simple quadratic equation.

]]>

In the centre we have the well-known quadratic solution.

The ± symbol comes from the top row of the Arithmetic palette.

To the right the user typed x:= and SMath gave both values of the x variable in vector form.

On the last line we see how to refer to an individual root (x1 and x2); used here to check that ax² +bx +c

does in fact equal zero with each x-value: clearly x1 = 3 and x2 = –6.

]]>

x 2^3 x * + 18 - 0 ≡

a 1 :

b 3 :

c 18 - :

x b - b 2^4 a * c * - sqrt ± 2 a * / :

x 36 - 21 sys

But in some circumstances there is analytical solution and we must rely on numerical approximations.

SMath has three functions for these operations: roots, polyroots and solve.

In addition there is a tool within the Calculations menu for solving equations.

Below is an example of the Solve tool from the Calculations menu.

]]>

Referring to the figure above, the user entered the line of text and the next three lines.

Carefully note that the entry f(x, b, c)=0 has a Boolean equality symbol inserted using the Boolean palette.

Such symbols appear bold on the page.

Then with the x with the function selected (hence is shows in blue box), the Calculation menu was opened and

Solve was clicked. SMath responded by inserting the two valued vector with the solutions to the equation.

Note how the results are given to the full 15 digit precision notwithstanding that in Tools | Options decimal places

were set to 4.

]]>

a b c f a 2^2 b * - 2 c + / 2 - :

b 1 :

c 2 :

a b c f 0 ≡

3.16227766503521 - 3.16227758007117 21 mat

The roots function has two forms; here are simplified syntaxes for each: (a) roots(function, variable) and

(b) roots(function, variable, guess). We shall see that, whenever possible, it is advisable to provide a value

for guess – to use the 3-argument form rather than the 2-argument form.

The guess value tells SMath were to start seeking a solution.

In the example below we find the two roots of e^x + x² -5x =0. The graph indicates roots at about 0.25 and 1.75.

The expression root1:=roots(f(x),x) returns the first value. When we give a value to guess, as in the next two

calls to roots, we can control which root gets returned. Note that the roots() results do not make f(x) equal

exactly zero but results that do so close to the precision of the system.

]]>

establish graph to get starting points

]]>

x f e x^x 2^+ 5 x * - :

u f e u^u 2^+ 5 u * - :

xx f u f x f

root2 u f u roots : 0.2805

root3 u f u roots : 0.2805

root4 u f u 0 roots : 0.2805

root5 u f u 4 roots : 1.734

root2 f root5 f 21 mat 4.4409 1015 -^* - 4.6185 1014 -^* - 21 mat