[PDF] Maxima by Example: Ch4: Solving Equations ∗

4 1 3 General Quadratic Equation or Function linsolve solves a system of simultaneous linear equations for the speci ed variables and returns a list of the solutions allroots nds all the real and complex roots of a real univariate polynomial

3 1 Solving Linear Equations 2 3 2 Solving Quadratic Equations 13 3 3 Solving Polynomial Equations 31 3 4 Solving Simultaneous Linear Equations 42

[PDF] Equations Linear, Quadratic, Cubic and Higher Orders

to solve simultaneous linear and quadratic equations • nature of roots The higher degree equations are also called higher degree polynomials or polynomial

[PDF] Module M14 Solving equations

Simultaneous linear equations 2 4 Graphical In Section 3 the discussion is extended to the solution of quadratic equations by Section 4 graphical and numerical procedures for solving polynomial equations are described The first thing

[PDF] Unit 4 Polynomials and Quadratic Functions Algebra - Long Beach

23 jui 2017 · Factoring polynomials breaks apart more complex expressions into smaller, more easily Systems consisting of a linear equation and a quadratic equation can be solved Using algebraic methods, such as substitution and

[PDF] Maxima by Example: Ch4: Solving Equations ∗ - CSULB

6 Polynomial Equations

ination could be adapted to simultaneous polynomial equations in two or be found by solving a series of linear or quadratic equations, which is why they

[PDF] Chapter One

1 jui 2016 · Objective 1: Solving Higher Order Polynomial Equations Linear equations and quadratic equations are both examples of polynomial equations of first and second degree, quadratic equation by using a substitution

[PDF] fx-991EX/fx-570EX - CASIO

SIMULTANEOUS EQUATIONS From the Main Menu, use To solve the following system of simultaneous equations , select 1(Simul To start solving polynomial equations, in the Equation/Func icon, the quadratic template, and press p

[PDF] Maxima by Example: Ch4: Solving Equations ∗ - CSULB

4.1.1 The Maxima Function solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1.2 solve with Expressions or Functions & the multiplicities List . . . . . . . . . . . . . . . . . . . . 4

4.1.3 General Quadratic Equation or Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.1.4 Checking Solutions with subst or ev and a "Do Loop" . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.5 The One Argument Form of solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.6 Using disp, display, and print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.7 Checking Solutions using map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.8 Psuedo-PostFix Code: %% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.9 Using an Expression Rather than a Function with Solve . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.10 Escape Speed from the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.11 Cubic Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.12 Trigonometric Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.13 Equation or Expression Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 15

4.2 One Equation Numerical Solutions: allroots, realroots, nd

root . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Comparison of realroots with allroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.2 Intersection Points of Two Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.3 Transcendental Equations and Roots: nd

root . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.4 nd

root: Quote that Function! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.5 newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Two or More Equations: Symbolic and Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Numerical or Symbolic Linear Equations with solve or linsolve . . . . . . . . . . . . . . . . . . 28

4.3.2 Matrix Methods for Linear Equation Sets: linsolve

lu . . . . . . . . . . . . . . . . . . . . . . 29

4.3.3 Symbolic Linear Equation Solutions: Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.4 Multiple Solutions from Multiple Right Hand Sides . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.5 Three Linear Equation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.6 Surpressing rat Messages: ratprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.7 Non-Linear Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.8 General Sets of Nonlinear Equations: eliminate, mnewton . . . . . . . . . . . . . . . . . . . . . 37

4.3.9 Intersections of Two Circles: implicit

plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.10 Using Draw for Implicit Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.11 Another Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.12 Error Messages and Do It Yourself Mnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.13 Automated Code for mymnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

This document is part of a series of notes titled "Maxima by Example" and is made available NON-PROFIT PRINTING AND DISTRIBUTION IS PERMITTED.

You may make copies of this document and distribute them to others as long as you charge no more than the

costs of printing. These notes (with some modications) will be published in book form eventually via Lulu.com in an ar- a low cost paperbound version of these notes. This chapter gives examples of the following Maxima functions: We also encourage the use of two dimensional plots to approximately locate solutions.

This chapter does not yet include "Solving Recurrence Relations", and "Solving One Hundred Equations".

Let's use conservation of mechanical energy (kinetic plus potential) to rst calculate the initial radial speed

a rocket must have near the surface of the earth to achieve a nal required radial speed far from the earth (far

We can make a simple plot of our expression to see the periodic behavior and the approximate location of

the real roots.-4 -6-4-2 0 2 4 6 x0.0 sin(x)2-2*sin(x)-3

Figure 1: plot of ex3

(%i1) 2 *(a*log(x) + 2*a*log(y))$ (%i2) logcontract(%); 2 4 (%o2) a log(x y ) root x2¡x¡1 -10

0 1 2 3 4 5

xhx rx=hx - kx

Figure 2: Intersection Points are Zeroes of rx

Here is code you can use to make something close to the above plot.

To nd the roots of transcendental expressions, for example, we can rst make a plot of the expression, and

then use??? root??? ?? ?? root error root abs root rel -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6

0 0.2 0.4 0.6 0.8 1

plot of x - cos(x)

Figure 3: Plot of x - cos(x)

We can use either an expression or function as the entry to the rst slot of??? root? ? ??? ???? ??? ???? f(x) = cos(x=¼)e¡(x=4)2¡sin(x3=2)¡5=4 ???? ?? ??? ???? ??f(x)?-2.5 -1.5 -0.5 0.5

0 1 2 3 4 5

plot of f(x)

Figure 4: Plot of f(x)

root: Quote that Function! ???root?? ??? root?? ???

The important thing to stress is that the???

2y dy?

root?

The following methods succeed.

i+1=xi¡f(xi)

0(xi):

ax+by=c;dx+ey=f A simple numerical (rather than symbolic) two equation example: lu? If we re-cast the two equation problem we have just been solving in the form of a matrix equation

One should always check solutions when using computer algebra software, since the are occasional bugs in

ax+by=c; dx+ey=f; y=¡x=3: plot plot??? -3-2-1 0 1 2 3 4 5 6

Intersections of Two Circles

Figure 5: two circles

-0.5 0.5 1.5 -3-2-1 0 1 2 3 dennis and schnabel example

Figure 6: Dennis and Schnabel Example

root. We rewrite the equations as expressions here.quotesdbs_dbs7.pdfusesText_5

[PDF] [PDF] Maxima by Example: Ch4: Solving Equations ∗ - CSULB

4.1.1 The Maxima Function solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1.2 solve with Expressions or Functions & the multiplicities List . . . . . . . . . . . . . . . . . . . . 4

4.1.3 General Quadratic Equation or Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.1.4 Checking Solutions with subst or ev and a "Do Loop" . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.5 The One Argument Form of solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.6 Using disp, display, and print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.7 Checking Solutions using map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1.8 Psuedo-PostFix Code: %% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.9 Using an Expression Rather than a Function with Solve . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.10 Escape Speed from the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1.11 Cubic Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.12 Trigonometric Equation or Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.13 Equation or Expression Containing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 15

4.2 One Equation Numerical Solutions: allroots, realroots, nd

4.2.1 Comparison of realroots with allroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.2 Intersection Points of Two Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.3 Transcendental Equations and Roots: nd

4.2.4 nd

4.2.5 newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Two or More Equations: Symbolic and Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Numerical or Symbolic Linear Equations with solve or linsolve . . . . . . . . . . . . . . . . . . 28

4.3.2 Matrix Methods for Linear Equation Sets: linsolve

4.3.3 Symbolic Linear Equation Solutions: Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.4 Multiple Solutions from Multiple Right Hand Sides . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.5 Three Linear Equation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.6 Surpressing rat Messages: ratprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.7 Non-Linear Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.8 General Sets of Nonlinear Equations: eliminate, mnewton . . . . . . . . . . . . . . . . . . . . . 37

4.3.9 Intersections of Two Circles: implicit

4.3.10 Using Draw for Implicit Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.11 Another Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.12 Error Messages and Do It Yourself Mnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.13 Automated Code for mymnewton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 1: plot of ex3

0 1 2 3 4 5

Figure 2: Intersection Points are Zeroes of rx

0 0.2 0.4 0.6 0.8 1

Figure 3: Plot of x - cos(x)

0 1 2 3 4 5

Figure 4: Plot of f(x)

The important thing to stress is that the???

2y dy?

The following methods succeed.

0(xi):

Intersections of Two Circles

Figure 5: two circles

Figure 6: Dennis and Schnabel Example