Dept. of Theoretical and Applied Mechanics, Cornell University

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

To invoke MAXIMA in Linux, type

maxima<ret>

The computer will display a greeting of the sort:

GCL (GNU Common Lisp) Version(2.3) ter jun 27 14:16:29 BRT 2000 Licensed under GNU Library General Public License Contains Enhancements by W. Schelter Maxima 5.4 ter jun 27 14:16:11 BRT 2000 (with enhancements by W. Schelter). Licensed under the GNU Public License (see file COPYING) (C1)

The `(C1)` 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. `C` labels denote your commands and `D` labels
denote Displays of the machine's response. *Never use variable names like C1 or
D5, as these will be confused with the lines so labeled*.

MAXIMA is pragmatic about lower and upper case: regardless of your typing `sin(x)` or `
SIN(x)`,
` %e^x` or

- To end a MAXIMA session, type
`quit();`. If you type`^C`, here is what happens:Correctable error: Console interrupt. Signalled by MACSYMA-TOP-LEVEL. If continued: Type :r to resume execution, or :q to quit to top level. Broken at SYSTEM:TERMINAL-INTERRUPT. Type :H for Help. MAXIMA>>:q (C1)

Notice that typing`:q`or`:t`(for*top level*) after the`MAXIMA>>`prompt gets you back to the MAXIMA level.`^Y`, on the other hand, won't have any effect but being echoed on the screen; finally`^Z`will have the same effect as`quit();`. (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.) - To abort a computation without leaving MAXIMA, type
`^C`. It is important for you to know how to do this in case, for example, you begin a computation which is taking too long. Remember to type`:q`at the`MAXIMA>>`prompt to return to MAXIMA. For example:(C1) sum(1/x^2,x,1,1000); Correctable error: Console interrupt. Signalled by MACSYMA-TOP-LEVEL. If continued: Type :r to resume execution, or :q to quit to top level. Broken at SYSTEM:TERMINAL-INTERRUPT. Type :H for Help. MAXIMA>>:q (C2)

- 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, supresses 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
`(C5)`, you may do so without typing it over again by preceding its label with two single quotes (`''`), i.e.,`''C5`. (Note that simply inputing`C5`will not do the job - try it.) - If you want to refer to the immediately preceding result computed my MAXIMA, you can either
use its
`D`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`(or`%E`),`%i`(or`%I`), and`%pi`(or`%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.

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. E.g.,

(C1) 1/100+1/101; 201 (D1) ----- 10100If irrational numbers are involved in a computation, they are kept in symbolic form:

(C2) (1+sqrt(2))^5; 5 (D2) (SQRT(2) + 1) (C3) expand(%); (D3) 29 SQRT(2) + 41However, it is often useful to express a result in decimal notation. This may be accomplished by following the expression you want expanded by ``

(C4) %,numer; (D4) 82.01219330881976Note the use here of

(C5) bfloat(d3); (D5) 8.201219330881976B1The number of significant figures displayed is controlled by the MAXIMA variable

(C6) fpprec; (D6) 16Here we reset

(C7) fpprec:100; (D7) 100 (C8) ''c5; (D8) 8.20121933088197564152489730020812442785204843859314941221237124017312418# 7540110412666123849550160561B1Note the use of two single quotes (

(C9) 100!; (D9) 9332621544394415268169923885626670049071596826438162146859296389521759999# 322991560894146397615651828625369792082722375825118521091686400000000000000000# 0000000

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:

(C1) (x+3*y+x^2*y)^3; 2 3 (D1) (x y + 3 y + x) (C2) expand(%); 6 3 4 3 2 3 3 5 2 3 2 2 4 (D2) x y + 9 x y + 27 x y + 27 y + 3 x y + 18 x y + 27 x y + 3 x y 2 3 + 9 x y + xNow suppose we wanted to substitute

(C3) d2,x=5/z; 2 3 2 3 2 135 y 675 y 225 y 2250 y 125 5625 y 1875 y 9375 y (D3) ------ + ------ + ----- + ------- + --- + ------- + ------ + ------- z 2 2 3 3 4 4 5 z z z z z z z 3 15625 y 3 + -------- + 27 y 6 z |

The MAXIMA function RATSIMP will place this over a common denominator:

(C4) ratsimp(%); 3 6 2 5 3 4 2 3 (D4) (27 y z + 135 y z + (675 y + 225 y) z + (2250 y + 125) z 3 2 2 3 6 + (5625 y + 1875 y) z + 9375 y z + 15625 y )/zExpressions may also be

(C5) factor(%); 2 3 (3 y z + 5 z + 25 y) (D5) ---------------------- 6 zMAXIMA can obtain exact solutions to systems of nonlinear algebraic equations. In this example we

(C6) a + b*c=1; (D6) b C + a = 1 (C7) b - a*c=0; (D7) b - a C = 0 (C8) a+b=5; (D8) b + a = 5 (C9) solve([d6,d7,d8],[a,b,c]); 25 SQRT(79) %I + 25 5 SQRT(79) %I + 5 SQRT(79) %I + 1 (D9) [[a = -------------------, b = -----------------, C = ---------------], 6 SQRT(79) %I - 34 SQRT(79) %I + 11 10 25 SQRT(79) %I - 25 5 SQRT(79) %I - 5 SQRT(79) %I - 1 [a = -------------------, b = -----------------, C = - ---------------]] 6 SQRT(79) %I + 34 SQRT(79) %I - 11 10Note that the display consists of a ``list'', i.e., some expression contained between two brackets

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:

(C10) sin(u+v)*cos(u)^3; 3 (D10) COS (u) SIN(v + u) (C11) trigexpand(%); 3 (D11) COS (u) (COS(u) SIN(v) + SIN(u) COS(v))The function

(C12) trigreduce(d10); SIN(v + 4 u) + SIN(v - 2 u) 3 SIN(v + 2 u) + 3 SIN(v) (D12) --------------------------- + ------------------------- 8 8The functions

(C13) w:3+k*%i; (D13) %I k + 3 (C14) w^2*%e^w; 2 %I k + 3 (D14) (%I k + 3) %E (C15) realpart(%); 3 2 3 (D15) %E (9 - k ) COS(k) - 6 %E k SIN(k)

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`:

(C1) f:x^3*%E^(k*x)*sin(w*x); 3 k x (D1) x %E SIN(w x)We compute the derivative of

(C2) diff(f,x); 3 k x 2 k x 3 k x (D2) k x %E SIN(w x) + 3 x %E SIN(w x) + w x %E COS(w x)Now we find the indefinite integral of

(C3) integrate(f,x); 6 3 4 5 2 7 3 (D3) (((k w + 3 k w + 3 k w + k ) x 6 2 4 4 2 6 2 4 3 2 5 + (3 w + 3 k w - 3 k w - 3 k ) x + (- 18 k w - 12 k w + 6 k ) x 4 2 2 4 k x - 6 w + 36 k w - 6 k ) %E SIN(w x) 7 2 5 4 3 6 3 5 3 3 5 2 + ((- w - 3 k w - 3 k w - k w) x + (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 COS(w x)) 8 2 6 4 4 6 2 8 /(w + 4 k w + 6 k w + 4 k w + k )A slight change in syntax gives definite integrals:

(C4) integrate(1/x^2,x,1,inf); (D4) 1 (C5) integrate(1/x,x,0,inf); Integral is divergent -- an error. Quitting. To debug this try DEBUGMODE(TRUE);)Next we define the simbol

(C6) g:f/sinh(k*x)^4; 3 k x x %E SIN(w x) (D6) ----------------- 4 SINH (k x) (C7) taylor(g,x,0,3); 2 3 2 2 3 3 w w x (w k + w ) x (3 w k + w ) x (D7)/T/ -- + --- - -------------- - ---------------- + . . . 4 3 4 3 k k 6 k 6 k |

The limit of

(C8) limit(g,x,0); w (D8) -- 4 kMAXIMA also permits derivatives to be represented in unevaluated form (note the quote):

(C9) 'diff(y,x); dy (D9) -- dxThe quote operator in

(C10) diff(y,x); (D10) 0Using the quote operator we can write differential equations:

(C11) 'diff(y,x,2) + 'diff(y,x) + y; 2 d y dy (D11) --- + -- + y 2 dx dxMAXIMA's

(C12) ode2(d11,y,x); - x/2 SQRT(3) x SQRT(3) x (D12) y = %E (%K1 SIN(---------) + %K2 COS(---------)) 2 2

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:

(C1) 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 ] [ ] (D1) [ 1 0 1 ] [ ] [ 1 1 0 ]Next we find its transpose, determinant and inverse:

(C2) transpose(m); [ 0 1 1 ] [ ] (D2) [ 1 0 1 ] [ ] [ a 1 0 ] (C3) determinant(m); (D3) a + 1 (C4) invert(m),detout; [ - 1 a 1 ] [ ] [ 1 - a a ] [ ] [ 1 1 - 1 ] (D4) ----------------- a + 1In

(C5) m.d4; [ - 1 a 1 ] [ ] [ 1 - a a ] [ 0 1 a ] [ ] [ ] [ 1 1 - 1 ] (D5) [ 1 0 1 ] . ----------------- [ ] a + 1 [ 1 1 0 ] (C6) expand(%); [ a 1 ] [ ----- + ----- 0 0 ] [ a + 1 a + 1 ] [ ] [ a 1 ] (D6) [ 0 ----- + ----- 0 ] [ a + 1 a + 1 ] [ ] [ a 1 ] [ 0 0 ----- + ----- ] [ a + 1 a + 1 ] (C7) factor(%); [ 1 0 0 ] [ ] (D7) [ 0 1 0 ] [ ] [ 0 0 1 ]In order to find the eigenvalues and eigenvectors of

(C8) eigenvectors(m); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS SQRT(4 a + 5) - 1 SQRT(4 a + 5) + 1 (D8) [[[- -----------------, -----------------, - 1], [1, 1, 1]], 2 2 SQRT(4 a + 5) - 1 SQRT(4 a + 5) - 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 + 2In D8, 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:
(C9) part(%,2); SQRT(4 a + 5) - 1 SQRT(4 a + 5) - 1 (D9) [1, - -----------------, - -----------------] 2 a + 2 2 a + 2 |

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 `f _{x}` and

/* -------------------------------------------------------------------------- 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

(C1) batch("critpts.max"); batching #/home/nldias/work/papers2000/intromax/critpts.max (C2) (C2) 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)) (C3) 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 (D3) [[x = 0.4588955685487 %I + 0.35897908710869, y = 0.49420173682751 %I - 0.12257873677837], [x = 0.35897908710869 - 0.4588955685487 %I, y = - 0.49420173682751 %I - 0.12257873677837], [x = 0.41875423272348 %I - 0.69231242044203, y = 0.4559120701117 - 0.86972626928141 %I], [x = - 0.41875423272348 %I - 0.69231242044203, y = 0.86972626928141 %I + 0.4559120701117]]

See the MAXIMA Manual in the `maxima-5.4/info/` directory in texinfo or html format. 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`NUMER`ical 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 BATCH 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
`B1`th part of`A`, then the`B2`th 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`SAVE`d 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 MACSYMA'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)`. MAXIMA also supports Taylor expansions in more than one independent variable; see the Manual for details.^{D} `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

^{1} Adapted from ``Perturbation Methods, Bifurcation Theory and Computer Algebra'' by Rand and
Armbruster, Springer, 1987

^{2} Adapted to L^{A}T_{E}X and HTML by Nelson L. Dias (nldias@simepar.br), SIMEPAR Technological Institute and Federal University of Paranį, Brazil

File translated from T

On 10 Jul 2000, 18:41.