# Interface to GNU Octave¶

GNU Octave is a free software (GPL) MATLAB-like program with numerical routines for integrating, solving systems of equations, special functions, and solving (numerically) differential equations. Please see http://octave.org/ for more details.

The commands in this section only work if you have the optional “octave” interpreter installed and available in your PATH. It’s not necessary to install any special Sage packages.

EXAMPLES:

sage: octave.eval('2+2')    # optional - octave
'ans = 4'

sage: a = octave(10)        # optional - octave
sage: a**10                 # optional - octave
1e+10


LOG: - creation (William Stein) - ? (David Joyner, 2005-12-18) - Examples (David Joyner, 2005-01-03)

## Computation of Special Functions¶

Octave implements computation of the following special functions (see the maxima and gp interfaces for even more special functions):

airy
Airy functions of the first and second kind, and their derivatives.
airy(0,x) = Ai(x), airy(1,x) = Ai'(x), airy(2,x) = Bi(x), airy(3,x) = Bi'(x)
besselj
Bessel functions of the first kind.
bessely
Bessel functions of the second kind.
besseli
Modified Bessel functions of the first kind.
besselk
Modified Bessel functions of the second kind.
besselh
Compute Hankel functions of the first (k = 1) or second (k = 2) kind.
beta
The Beta function,
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
betainc
The incomplete Beta function,
erf
The error function,
erfinv
The inverse of the error function.
gamma
The Gamma function,
gammainc
The incomplete gamma function,

For example,

sage: octave("airy(3,2)")         # optional - octave
4.10068
sage: octave("beta(2,2)")         # optional - octave
0.166667
sage: octave("betainc(0.2,2,2)")  # optional - octave
0.104
sage: octave("besselh(0,2)")      # optional - octave
(0.223891,0.510376)
sage: octave("besselh(0,1)")      # optional - octave
(0.765198,0.088257)
sage: octave("besseli(1,2)")      # optional - octave
1.59064
sage: octave("besselj(1,2)")      # optional - octave
0.576725
sage: octave("besselk(1,2)")      # optional - octave
0.139866
sage: octave("erf(0)")            # optional - octave
0
sage: octave("erf(1)")            # optional - octave
0.842701
sage: octave("erfinv(0.842)")     # optional - octave
0.998315
sage: octave("gamma(1.5)")        # optional - octave
0.886227
sage: octave("gammainc(1.5,1)")   # optional - octave
0.77687


The Octave interface reads in even very long input (using files) in a robust manner:

sage: t = '"%s"'%10^10000   # ten thousand character string.
sage: a = octave.eval(t + ';')    # optional - octave, < 1/100th of a second
sage: a = octave(t)               # optional - octave


Note that actually reading a back out takes forever. This must be fixed ASAP - see http://trac.sagemath.org/sage_trac/ticket/940/.

## Tutorial¶

EXAMPLES:

sage: octave('4+10')              # optional - octave
14
sage: octave('date')              # optional - octave; random output
18-Oct-2007
sage: octave('5*10 + 6')          # optional - octave
56
sage: octave('(6+6)/3')           # optional - octave
4
sage: octave('9')^2               # optional - octave
81
sage: a = octave(10); b = octave(20); c = octave(30)    # optional - octave
sage: avg = (a+b+c)/3             # optional - octave
sage: avg                         # optional - octave
20
sage: parent(avg)                 # optional - octave
Octave

sage: my_scalar = octave('3.1415')       # optional - octave
sage: my_scalar                          # optional - octave
3.1415
sage: my_vector1 = octave('[1,5,7]')     # optional - octave
sage: my_vector1                         # optional - octave
1     5     7
sage: my_vector2 = octave('[1;5;7]')     # optional - octave
sage: my_vector2                         # optional - octave
1
5
7
sage: my_vector1 * my_vector2            # optional - octave
75

class sage.interfaces.octave.Octave(maxread=100, script_subdirectory='', logfile=None, server=None, server_tmpdir=None)

Interface to the Octave interpreter.

EXAMPLES:

sage: octave.eval("a = [ 1, 1, 2; 3, 5, 8; 13, 21, 33 ]")    # optional - octave
'a =\n\n 1 1 2\n 3 5 8\n 13 21 33\n\n'
sage: octave.eval("b = [ 1; 3; 13]")                         # optional - octave
'b =\n\n 1\n 3\n 13\n\n'
sage: octave.eval("c=a \\ b") # solves linear equation: a*c = b  # optional - octave; random output
'c =\n\n 1\n 7.21645e-16\n -7.21645e-16\n\n'
sage: octave.eval("c")                                 # optional - octave; random output
'c =\n\n 1\n 7.21645e-16\n -7.21645e-16\n\n'

clear(var)

Clear the variable named var.

EXAMPLES:

sage: octave.set('x', '2') # optional - octave
sage: octave.clear('x') # optional - octave
sage: octave.get('x') # optional - octave
"error: x' undefined near line ... column 1"

console()

Spawn a new Octave command-line session.

This requires that the optional octave program be installed and in your PATH, but no optional Sage packages need be installed.

EXAMPLES:

sage: octave_console()         # not tested
GNU Octave, version 2.1.73 (i386-apple-darwin8.5.3).
Copyright (C) 2006 John W. Eaton.
...
octave:1> 2+3
ans = 5
octave:2> [ctl-d]


Pressing ctrl-d exits the octave console and returns you to Sage. octave, like Sage, remembers its history from one session to another.

de_system_plot(f, ics, trange)

Plots (using octave’s interface to gnuplot) the solution to a $$2\times 2$$ system of differential equations.

INPUT:

• f - a pair of strings representing the differential equations; The independent variable must be called x and the dependent variable must be called y.
• ics - a pair [x0,y0] such that x(t0) = x0, y(t0) = y0
• trange - a pair [t0,t1]

OUTPUT: a gnuplot window appears

EXAMPLES:

sage: octave.de_system_plot(['x+y','x-y'], [1,-1], [0,2])  # not tested -- does this actually work (on OS X it fails for me -- William Stein, 2007-10)


This should yield the two plots $$(t,x(t)), (t,y(t))$$ on the same graph (the $$t$$-axis is the horizontal axis) of the system of ODEs

$\begin{split}x' = x+y, x(0) = 1;\qquad y' = x-y, y(0) = -1, \quad\text{for}\quad 0 < t < 2.\end{split}$
get(var)

Get the value of the variable var.

EXAMPLES:

sage: octave.set('x', '2') # optional - octave
sage: octave.get('x') # optional - octave
' 2'

quit(verbose=False)

EXAMPLES:

sage: o = Octave()
sage: o._start()    # optional - octave
sage: o.quit(True)  # optional - octave
Exiting spawned Octave process.

sage2octave_matrix_string(A)

Return an octave matrix from a Sage matrix.

INPUT: A Sage matrix with entries in the rationals or reals.

OUTPUT: A string that evaluates to an Octave matrix.

EXAMPLES:

sage: M33 = MatrixSpace(QQ,3,3)
sage: A = M33([1,2,3,4,5,6,7,8,0])
sage: octave.sage2octave_matrix_string(A)   # optional - octave
'[1, 2, 3; 4, 5, 6; 7, 8, 0]'


AUTHORS:

• David Joyner and William Stein
set(var, value)

Set the variable var to the given value.

EXAMPLES:

sage: octave.set('x', '2') # optional - octave
sage: octave.get('x') # optional - octave
' 2'

solve_linear_system(A, b)

Use octave to compute a solution x to A*x = b, as a list.

INPUT:

• A - mxn matrix A with entries in QQ or RR
• b - m-vector b entries in QQ or RR (resp)

OUTPUT: An list x (if it exists) which solves M*x = b

EXAMPLES:

sage: M33 = MatrixSpace(QQ,3,3)
sage: A   = M33([1,2,3,4,5,6,7,8,0])
sage: V3  = VectorSpace(QQ,3)
sage: b   = V3([1,2,3])
sage: octave.solve_linear_system(A,b)    # optional - octave (and output is slightly random in low order bits)
[-0.33333299999999999, 0.66666700000000001, -3.5236600000000002e-18]


AUTHORS:

• David Joyner and William Stein
version()

Return the version of Octave.

OUTPUT: string

EXAMPLES:

sage: octave.version()   # optional - octave; random output depending on version
'2.1.73'

class sage.interfaces.octave.OctaveElement(parent, value, is_name=False, name=None)
sage.interfaces.octave.octave_console()

Spawn a new Octave command-line session.

This requires that the optional octave program be installed and in your PATH, but no optional Sage packages need be installed.

EXAMPLES:

sage: octave_console()         # not tested
GNU Octave, version 2.1.73 (i386-apple-darwin8.5.3).
Copyright (C) 2006 John W. Eaton.
...
octave:1> 2+3
ans = 5
octave:2> [ctl-d]


Pressing ctrl-d exits the octave console and returns you to Sage. octave, like Sage, remembers its history from one session to another.

sage.interfaces.octave.octave_version()

Return the version of Octave installed.

EXAMPLES:

sage: octave_version()    # optional - octave; and output is random
'2.9.12'


EXAMPLES:

sage: from sage.interfaces.octave import reduce_load_Octave
Octave
`