Introduction to Maxima
Richard H. Rand
Dept. of Theoretical and Applied Mechanics, Cornell University
1
Copyright (c) 1988-2010 Richard H. Rand.
This document is free; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free
Software Foundation. See the GNU General Public License for more
details at http://www.gnu.org/copyleft/gpl.html
Contents
1 Introduction
2 Special keys and symbols
3 Arithmetic
4 Algebra
5 Calculus
6 Matrix calculations
7 Programming in Maxima
8 A partial list of Maxima functions
1 Introduction
To invoke Maxima in a console, type
maxima <enter>
The computer will display a greeting of the sort:
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1)
The (%i1) is a "label". Each input or output line is
labelled and can be referred to by its own label for the rest of the
session. i labels denote your commands and o labels
denote displays of the machine's response. Never use variable
names like %i1 or %o5, as these will be confused with
the lines so labeled.
Maxima distinguishes lower and upper case. All built-in functions
have names which are lowercase only (sin, cos, save,
load, etc). Built-in constants have lowercase names (%e,
%pi, inf, etc). If you type SIN(x) or
Sin(x), Maxima assumes you mean something other than the built-in
sin function. User-defined functions and variables can have
names which are lower or upper case or both. foo(XY),
Foo(Xy), FOO(xy) are all different.
2 Special keys and symbols
- To end a Maxima session, type quit();.
- To abort a computation without leaving Maxima, type ^C.
(Here ^ stands for the control key, so
that ^C means first press the key marked control and hold it down while pressing the C key.)
It is important for you to
know how to do this in case, for example, you begin a computation which is taking too long.
For example:
(%i1) sum (1/x^2, x, 1, 100000)$
^C
Maxima encountered a Lisp error:
Interactive interrupt at #x7FFFF74A43C3.
Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.
(%i2)
- In order to tell Maxima that you have finished your command, use
the semicolon (;), followed by a return. Note that the return
key alone does not signal that you are done with your input.
- An alternative input terminator to the semicolon (;) is
the dollar sign ($), which, however, suppresses the display of
Maxima's computation. This is useful if you are computing some long
intermediate result, and you don't want to waste time having it
displayed on the screen.
- If you wish to repeat a command which you have already given,
say on line (%i5), you may do so without typing it over again
by preceding its label with two single quotes ("), i.e.,
"%i5. (Note that simply inputing %i5 will not do the job
- try it.)
- If you want to refer to the immediately preceding result
computed by Maxima, you can either use its o label, or you can
use the special symbol percent (%).
- The standard quantities e (natural log base), i (square root
of -1) and p (3.14159¼) are respectively referred to as
%e, %i,
and %pi. Note that the use of % here as a prefix
is completely unrelated to the use of % to refer to the
preceding result computed.
- In order to assign a value to a variable, Maxima uses the colon
(:), not the equal sign. The equal sign is used for
representing equations.
3 Arithmetic
The common arithmetic operations are
- +
- addition
- -
- subtraction
- *
- scalar multiplication
- /
- division
- ^
- or ** exponentiation
- .
- matrix multiplication
- sqrt(x)
- square root of x.
Maxima's output is characterized by exact (rational) arithmetic. For example,
(%i1) 1/100 + 1/101;
201
(%o1) -----
10100
If irrational numbers are involved in a computation, they are kept in symbolic form:
(%i2) (1 + sqrt(2))^5;
5
(%o2) (sqrt(2) + 1)
(%i3) expand (%);
7/2
(%o3) 3 2 + 5 sqrt(2) + 41
However, it is often useful to express a result in decimal notation.
This may be accomplished by following the expression you want expanded
by ",numer":
(%i4) %, numer;
(%o4) 82.01219330881977
Note the use here of %
to refer to the previous result. In this version of Maxima,
numer gives 16 significant figures, of which the last is often
unreliable. However, Maxima can offer arbitrarily high
precision by using the bfloat function:
(%i5) bfloat (%o3);
(%o5) 8.201219330881976b1
The number of significant figures displayed is controlled by the
Maxima variable fpprec, which has the default value of 16:
(%i6) fpprec;
(%o6) 16
Here we reset fpprec to yield 100 digits:
(%i7) fpprec: 100;
(%o7) 100
(%i8) ''%i5;
(%o8) 8.20121933088197564152489730020812442785204843859314941221\
2371240173124187540110412666123849550160561b1
Note the use of two single quotes (") in (%i8) to repeat
command (%i5). Maxima can handle very large numbers without
approximation:
(%i9) 100!;
(%o9) 9332621544394415268169923885626670049071596826438162146859\
2963895217599993229915608941463976156518286253697920827223758251\
185210916864000000000000000000000000
4 Algebra
Maxima's importance as a computer tool to facilitate analytical
calculations becomes more evident when we see how easily it does
algebra for us. Here's an example in which a polynomial is expanded:
(%i1) (x + 3*y + x^2*y)^3;
2 3
(%o1) (x y + 3 y + x)
(%i2) expand (%);
6 3 4 3 2 3 3 5 2 3 2
(%o2) x y + 9 x y + 27 x y + 27 y + 3 x y + 18 x y
2 4 2 3
+ 27 x y + 3 x y + 9 x y + x
Now suppose we wanted to substitute 5/z for x in the above
expression:
| |
(%i3) %o2, x=5/z;
2 3 2 3
135 y 675 y 225 y 2250 y 125 5625 y 1875 y
(%o3) ------ + ------ + ----- + ------- + --- + ------- + ------
z 2 2 3 3 4 4
z z z z z z
2 3
9375 y 15625 y 3
+ ------- + -------- + 27 y
5 6
z z
|
| |
The Maxima function ratsimp will place this over a common denominator:
(%i4) ratsimp (%);
3 6 2 5 3 4
(%o4) (27 y z + 135 y z + (675 y + 225 y) z
2 3 3 2 2
+ (2250 y + 125) z + (5625 y + 1875 y) z + 9375 y z
3 6
+ 15625 y )/z
Expressions may also be factored:
(%i5) factor (%);
2 3
(3 y z + 5 z + 25 y)
(%o5) ----------------------
6
z
Maxima can obtain exact solutions to systems of nonlinear algebraic
equations. In this example we solve three equations in the
three unknowns a, b, c:
(%i6) a + b*c = 1;
(%o6) b c + a = 1
(%i7) b - a*c = 0;
(%o7) b - a c = 0
(%i8) a + b = 5;
(%o8) b + a = 5
(%i9) solve ([%o6, %o7, %o8], [a, b, c]);
25 sqrt(79) %i + 25 5 sqrt(79) %i + 5
(%o9) [[a = -------------------, b = -----------------,
6 sqrt(79) %i - 34 sqrt(79) %i + 11
sqrt(79) %i + 1 25 sqrt(79) %i - 25
c = ---------------], [a = -------------------,
10 6 sqrt(79) %i + 34
5 sqrt(79) %i - 5 sqrt(79) %i - 1
b = -----------------, c = - ---------------]]
sqrt(79) %i - 11 10
Note that the display consists of a "list", i.e., some expression
contained between two brackets [ ... ], which itself contains
two lists. Each of the latter contain a distinct solution to the
simultaneous equations.
Trigonometric identities are easy to manipulate in Maxima. The
function trigexpand uses the sum-of-angles formulas to make the
argument inside each trig function as simple as possible:
(%i10) sin(u + v) * cos(u)^3;
3
(%o10) cos (u) sin(v + u)
(%i11) trigexpand (%);
3
(%o11) cos (u) (cos(u) sin(v) + sin(u) cos(v))
The function trigreduce, on the other hand, converts an
expression into a form which is a sum of terms, each of which contains
only a single sin or cos:
(%i12) trigreduce (%o10);
sin(v + 4 u) + sin(v - 2 u) 3 sin(v + 2 u) + 3 sin(v)
(%o12) --------------------------- + -------------------------
8 8
The functions realpart and imagpart will return the real
and imaginary parts of a complex expression:
(%i13) w: 3 + k*%i;
(%o13) %i k + 3
(%i14) w^2 * %e^w;
2 %i k + 3
(%o14) (%i k + 3) %e
(%i15) realpart (%);
3 2 3
(%o15) %e (9 - k ) cos(k) - 6 %e k sin(k)
5 Calculus
Maxima can compute derivatives and integrals, expand in Taylor series,
take limits, and obtain exact solutions to ordinary differential
equations. We begin by defining the symbol f to be the
following function of x:
(%i1) f: x^3 * %e^(k*x) * sin(w*x);
3 k x
(%o1) x %e sin(w x)
We compute the derivative of f with respect to x:
(%i2) diff (f, x);
3 k x 2 k x
(%o2) k x %e sin(w x) + 3 x %e sin(w x)
3 k x
+ w x %e cos(w x)
Now we find the indefinite integral of f with respect to x:
(%i3) integrate (f, x);
6 3 4 5 2 7 3
(%o3) (((k w + 3 k w + 3 k w + k ) x
6 2 4 4 2 6 2
+ (3 w + 3 k w - 3 k w - 3 k ) x
4 3 2 5 4 2 2 4
+ (- 18 k w - 12 k w + 6 k ) x - 6 w + 36 k w - 6 k )
k x 7 2 5 4 3 6 3
%e sin(w x) + ((- w - 3 k w - 3 k w - k w) x
5 3 3 5 2
+ (6 k w + 12 k w + 6 k w) x
5 2 3 4 3 3 k x
+ (6 w - 12 k w - 18 k w) x - 24 k w + 24 k w) %e
8 2 6 4 4 6 2 8
cos(w x))/(w + 4 k w + 6 k w + 4 k w + k )
A slight change in syntax gives definite integrals:
(%i4) integrate (1/x^2, x, 1, inf);
(%o4) 1
(%i5) integrate (1/x, x, 0, inf);
defint: integral is divergent.
-- an error. To debug this try: debugmode(true);
Next we define the simbol g in terms of f (previously
defined in %i1) and the hyperbolic sine function, and find its
Taylor series expansion (up to, say, order 3 terms) about the point
x = 0:
| |
(%i6) g: f / sinh(k*x)^4;
3 k x
x %e sin(w x)
(%o6) -----------------
4
sinh (k x)
(%i7) taylor (g, x, 0, 3);
2 3 2 2 3 3
w w x (w k + w ) x (3 w k + w ) x
(%o7)/T/ -- + --- - -------------- - ---------------- + . . .
4 3 4 3
k k 6 k 6 k
|
| |
The limit of g as x goes to 0 is computed as follows:
(%i8) limit (g, x, 0);
w
(%o8) --
4
k
Maxima also permits derivatives to be represented in unevaluated form
(note the quote):
(%i9) 'diff (y, x);
dy
(%o9) --
dx
The quote operator in (%i9) means "do not evaluate". Without
it, Maxima would have obtained 0:
(%i10) diff (y, x);
(%o10) 0
Using the quote operator we can write differential equations:
(%i11) 'diff (y, x, 2) + 'diff (y, x) + y;
2
d y dy
(%o11) --- + -- + y
2 dx
dx
Maxima's ode2 function can solve first and second order ODE's:
(%i12) ode2 (%o11, y, x);
- x/2 sqrt(3) x sqrt(3) x
(%o12) y = %e (%k1 sin(---------) + %k2 cos(---------))
2 2
6 Matrix calculations
Maxima can compute the determinant, inverse and eigenvalues and
eigenvectors of matrices which have symbolic elements (i.e., elements
which involve algebraic variables.) We begin by entering a matrix
m element by element:
(%i1) m: entermatrix (3, 3);
Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General
Answer 1, 2, 3 or 4 :
4;
Row 1 Column 1:
0;
Row 1 Column 2:
1;
Row 1 Column 3:
a;
Row 2 Column 1:
1;
Row 2 Column 2:
0;
Row 2 Column 3:
1;
Row 3 Column 1:
1;
Row 3 Column 2:
1;
Row 3 Column 3:
0;
Matrix entered.
[ 0 1 a ]
[ ]
(%o1) [ 1 0 1 ]
[ ]
[ 1 1 0 ]
Next we find its transpose, determinant and inverse:
(%i2) transpose (m);
[ 0 1 1 ]
[ ]
(%o2) [ 1 0 1 ]
[ ]
[ a 1 0 ]
(%i3) determinant (m);
(%o3) a + 1
(%i4) invert (m), detout;
[ - 1 a 1 ]
[ ]
[ 1 - a a ]
[ ]
[ 1 1 - 1 ]
(%o4) -----------------
a + 1
In (%i4), the modifier detout keeps the determinant
outside the inverse. As a check, we multiply m by its inverse
(note the use of the period to represent matrix multiplication):
(%i5) m . %o4;
[ - 1 a 1 ]
[ ]
[ 1 - a a ]
[ 0 1 a ] [ ]
[ ] [ 1 1 - 1 ]
(%o5) [ 1 0 1 ] . -----------------
[ ] a + 1
[ 1 1 0 ]
(%i6) expand (%);
[ a 1 ]
[ ----- + ----- 0 0 ]
[ a + 1 a + 1 ]
[ ]
[ a 1 ]
(%o6) [ 0 ----- + ----- 0 ]
[ a + 1 a + 1 ]
[ ]
[ a 1 ]
[ 0 0 ----- + ----- ]
[ a + 1 a + 1 ]
(%i7) factor (%);
[ 1 0 0 ]
[ ]
(%o7) [ 0 1 0 ]
[ ]
[ 0 0 1 ]
In order to find the eigenvalues and eigenvectors of m, we use the function
eigenvectors:
| |
(%i8) eigenvectors (m);
sqrt(4 a + 5) - 1 sqrt(4 a + 5) + 1
(%o8) [[[- -----------------, -----------------, - 1],
2 2
sqrt(4 a + 5) - 1 sqrt(4 a + 5) - 1
[1, 1, 1]], [[[1, - -----------------, - -----------------]],
2 a + 2 2 a + 2
sqrt(4 a + 5) + 1 sqrt(4 a + 5) + 1
[[1, -----------------, -----------------]], [[1, - 1, 0]]]]
2 a + 2 2 a + 2
In %o8, the first triple gives the eigenvalues of m and
the next gives their respective multiplicities (here each is
unrepeated). The next three triples give the corresponding
eigenvectors of m. In order to extract from this expression
one of these eigenvectors, we may use the part function:
(%i9) part (%o23, 2, 1, 1);
sqrt(4 a + 5) - 1 sqrt(4 a + 5) - 1
(%o9) [1, - -----------------, - -----------------]
2 a + 2 2 a + 2
|
| |
7 Programming in Maxima
So far, we have used Maxima in the interactive mode, rather like a
calculator. However, for computations which involve a repetitive
sequence of commands, it is better to execute a program. Here we
present a short sample program to calculate the critical points of a
function f of two variables x and y. The program
cues the user to enter the function f, then it computes the
partial derivatives fx and fy, and then it
uses the Maxima command solve to obtain solutions to
fx = fy = 0. The program is written outside of Maxima
with a text editor, and then loaded into Maxima with the batch
command. Here is the program listing:
/* --------------------------------------------------------------------------
this is file critpts.max:
as you can see, comments in maxima are like comments in C
Nelson Luis Dias, nldias@simepar.br
created 20000707
updated 20000707
--------------------------------------------------------------------------- */
critpts():=(
print("program to find critical points"),
/* ---------------------------------------------------------------------------
asks for a function
--------------------------------------------------------------------------- */
f:read("enter f(x,y)"),
/* ---------------------------------------------------------------------------
echoes it, to make sure
--------------------------------------------------------------------------- */
print("f = ",f),
/* ---------------------------------------------------------------------------
produces a list with the two partial derivatives of f
--------------------------------------------------------------------------- */
eqs:[diff(f,x),diff(f,y)],
/* ---------------------------------------------------------------------------
produces a list of unknowns
--------------------------------------------------------------------------- */
unk:[x,y],
/* ---------------------------------------------------------------------------
solves the system
--------------------------------------------------------------------------- */
solve(eqs,unk)
)$
The program (which is actually a function with no argument) is called
critpts. Each line is a valid Maxima command which could be
executed from the keyboard, and which is separated by the next command
by a comma. The partial derivatives are stored in a variable named
eqs, and the unknowns are stored in unk. Here is a sample
run:
(%i1) batch ("critpts.max");
batching #p/home/robert/tmp/maxima-clean/maxima/critpts.max
(%i2) critpts() := (print("program to find critical points"),
f : read("enter f(x,y)"), print("f = ", f),
eqs : [diff(f, x), diff(f, y)], unk : [x, y], solve(eqs, unk))
(%i3) critpts ();
program to find critical points
enter f(x,y)
%e^(x^3 + y^2)*(x + y);
2 3
y + x
f = (y + x) %e
(%o3) [[x = .4588955685487001 %i + .3589790871086935,
y = .4942017368275118 %i - .1225787367783657],
[x = .3589790871086935 - .4588955685487001 %i,
y = - .4942017368275118 %i - .1225787367783657],
[x = .4187542327234816 %i - .6923124204420268,
y = 0.455912070111699 - .8697262692814121 %i],
[x = - .4187542327234816 %i - .6923124204420268,
y = .8697262692814121 %i + 0.455912070111699]]
8 A partial list of Maxima functions
See the Maxima reference manual doc/html/maxima_toc.html (under
the main Maxima installation directory). From Maxima itself, you can
use describe(function name).
- allroots(a)
- Finds all the (generally complex) roots of
the polynomial equation A, and lists them in numerical
format (i.e. to 16 significant figures).
- append(a,b)
- Appends the list b to the list a,
resulting in a single list.
- batch(a)
- Loads and runs a program with filename a.
- coeff(a,b,c)
- Gives the coefficient of b raised to
the power c in expression a.
- concat(a,b)
- Creates the symbol ab.
- cons(a,b)
- Adds a to the list b as its first element.
- demoivre(a)
- Transforms all complex exponentials in
a to their trigonometric equivalents.
- denom(a)
- Gives the denominator of a.
- depends(a,b)
- Declares a to be a function of
b. This is useful for writing unevaluated derivatives, as in
specifying differential equations.
- desolve(a,b)
- Attempts to solve a linear system a of
ODE's for unknowns b using Laplace transforms.
- determinant(a)
- Returns the determinant of the square
matrix a.
- diff(a,b1,c1,b2,c2,...,bn,cn)
- Gives the mixed partial
derivative of a with respect to each bi, ci times.
For brevity, diff(a,b,1) may be represented by
diff(a,b). 'diff(...) represents the unevaluated
derivative, useful in specifying a differential equation.
- eigenvalues(a)
- Returns two lists, the first being the
eigenvalues of the square matrix a, and the second being their
respective multiplicities.
- eigenvectors(a)
- Does everything that eigenvalues
does, and adds a list of the eigenvectors of a.
- entermatrix(a,b)
- Cues the user to enter an a × b matrix, element by element.
- ev(a,b1,b2,...,bn)
- Evaluates a subject to the
conditions bi. In particular the bi may be equations,
lists of equations (such as that returned by solve), or
assignments, in which cases ev "plugs" the bi into
a. The Bi may also be words such as numer (in
which case the result is returned in numerical format), detout
(in which case any matrix inverses in a are performed with the
determinant factored out), or diff (in which case all
differentiations in a are evaluated, i.e., 'diff in
a is replaced by diff). For brevity in a manual command
(i.e., not inside a user-defined function), the ev may be
dropped, shortening the syntax to a,b1,b2,...,bn.
- expand(a)
- Algebraically expands a. In particular
multiplication is distributed over addition.
- exponentialize(a)
- Transforms all trigonometric functions
in a to their complex exponential equivalents.
- factor(a)
- Factors a.
- freeof(a,b)
- Is true if the variable a is not part
of the expression b.
- grind(a)
- Displays a variable or function a in a
compact format. When used with writefile and an editor
outside of Maxima, it offers a scheme for producing batch
files which include Maxima-generated expressions.
- ident(a)
- Returns an a × a
identity matrix.
- imagpart(a)
- Returns the imaginary part of a.
- integrate(a,b)
- Attempts to find the indefinite integral
of a with respect to b.
- integrate(a,b,c,d)
- Attempts to find the indefinite
integral of a with respect to b. taken from
b=c to b=d. The limits of integration c
and d may be taken is inf (positive infinity) of
minf (negative infinity).
- invert(a)
- Computes the inverse of the square matrix
a.
- kill(a)
- Removes the variable a with all its
assignments and properties from the current Maxima environment.
- limit(a,b,c)
- Gives the limit of expression a as
variable b approaches the value c. The latter may be
taken as inf of minf as in integrate.
- lhs(a)
- Gives the left-hand side of the equation a.
- loadfile(a)
- Loads a disk file with filename a from
the current default directory. The disk file must be in the proper
format (i.e. created by a save command).
- makelist(a,b,c,d)
- Creates a list of a's (each of
which presumably depends on b), concatenated from
b=c to b=d
- map(a,b)
- Maps the function a onto the
subexpressions of b.
- matrix(a1,a2,...,an)
- Creates a matrix consisting of the rows ai, where each
row ai is a list of m elements, [b1, b2, ..., bm].
- num(a)
- Gives the numerator of a.
- ode2(a,b,c)
- Attempts to solve the first- or second-order
ordinary differential equation a for b as a function of
c.
- part(a,b1,b2,...,bn)
- First takes the b1th part
of a, then the b2th part of that, and so on.
- playback(a)
- Displays the last a (an integer)
labels and their associated expressions. If a is omitted,
all lines are played back. See the Manual for other options.
- ratsimp(a)
- Simplifies a and returns a quotient of
two polynomials.
- realpart(a)
- Returns the real part of a.
- rhs(a)
- Gives the right-hand side of the equation a.
- save(a,b1,b2,..., bn)
- Creates a disk file with
filename a in the current default directory, of variables,
functions, or arrays bi. The format of the file permits it to
be reloaded into Maxima using the loadfile command.
Everything (including labels) may be saved by taking b1
equal to all.
- solve(a,b)
- Attempts to solve the algebraic equation
a for the unknown b. A list of solution equations is
returned. For brevity, if a is an equation of the form
c = 0, it may be abbreviated simply by the expression
c.
- string(a)
- Converts a to Maxima's linear notation
(similar to Fortran's) just as if it had been typed in and puts
a into the buffer for possible editing. The string'ed
expression should not be used in a computation.
- stringout(a,b1,b2,...,bn)
- Creates a disk file with
filename a in the current default directory, of variables
(e.g. labels) bi. The file is in a text format and is not
reloadable into Maxima. However the strungout expressions can be
incorporated into a Fortran, Basic or C program with a minimum of
editing.
- subst(a,b,c)
- Substitutes a for b in c.
- taylor(a,b,c,d)
- Expands a in a Taylor series in
b about b=c, up to and including the term
(b-c)d. Maxima also supports Taylor expansions in more
than one independent variable; see the Manual for details.
- transpose(a)
- Gives the transpose of the matrix a.
- trigexpand(a)
- Is a trig simplification function which
uses the sum-of-angles formulas to simplify the arguments of
individual sin's or cos's. For example,
trigexpand(sin(x+y)) gives cos(x) sin(y) + sin(x) cos(y).
- trigreduce(a)
- Is a trig simplification function which
uses trig identities to convert products and powers of sin and
cos into a sum of terms, each of which contains only a single
sin or cos. For example, trigreduce(sin(x)^2)
gives (1 - cos(2x))/2.
- trigsimp(a)
- Is a trig simplification function which
replaces tan, sec, etc., by their sin and
cos equivalents. It also uses the identity sin()2 + cos()2 = 1.
Footnotes:
1Adapted from "Perturbation Methods, Bifurcation Theory and Computer Algebra"
by Rand and Armbruster, Springer, 1987.
Adapted to LATEX and HTML by Nelson L. Dias (nldias@simepar.br),
SIMEPAR Technological Institute and Federal University of Paraná, Brazil.
Updated by Robert Dodier, August 2005.
File translated from
TEX
by
TTH,
version 3.85.
On 26 Apr 2010, 00:45.