ginsh(1) The GiNaC Group ginsh(1)
NAME
ginsh - GiNaC Interactive Shell
SYNPOSIS
ginsh [file...]
DESCRIPTION
ginsh is an interactive frontend for the GiNaC symbolic computation
framework. It is intended as a tool for testing and experimenting with
GiNaC's features, not as a replacement for traditional interactive com-
puter algebra systems. Although it can do many things these traditional
systems can do, ginsh provides no programming constructs like loops or
conditional expressions. If you need this functionality you are advised
to write your program in C++, using the "native" GiNaC class framework.
USAGE
INPUT FORMAT
After startup, ginsh displays a prompt ("> ") signifying that it is
ready to accept your input. Acceptable input are numeric or symbolic
expressions consisting of numbers (e.g. 42, 2/3 or 0.17), symbols
(e.g. x or result), mathematical operators like + and *, and func-
tions (e.g. sin or normal). Every input expression must be terminated
with either a semicolon (;) or a colon (:). If terminated with a semi-
colon, ginsh will evaluate the expression and print the result to std-
out. If terminated with a colon, ginsh will only evaluate the expres-
sion but not print the result. It is possible to enter multiple expres-
sions on one line. Whitespace (spaces, tabs, newlines) can be applied
freely between tokens. To quit ginsh, enter quit or exit, or type an
EOF (Ctrl-D) at the prompt.
COMMENTS
Anything following a double slash (//) up to the end of the line, and
all lines starting with a hash mark (#) are treated as a comment and
ignored.
NUMBERS
ginsh accepts numbers in the usual decimal notations. This includes ar-
bitrary precision integers and rationals as well as floating point num-
bers in standard or scientific notation (e.g. 1.2E6). The general
rule is that if a number contains a decimal point (.), it is an (inex-
act) floating point number; otherwise it is an (exact) integer or ra-
tional. Integers can be specified in binary, octal, hexadecimal or ar-
bitrary (2-36) base by prefixing them with #b, #o, #x, or #nR , respec-
tively.
SYMBOLS
Symbols are made up of a string of alphanumeric characters and the un-
derscore (_), with the first character being non-numeric. E.g. a and
mu_1 are acceptable symbol names, while 2pi is not. It is possible to
use symbols with the same names as functions (e.g. sin); ginsh is able
to distinguish between the two.
Symbols can be assigned values by entering
symbol = expression;
To unassign the value of an assigned symbol, type
unassign('symbol');
Assigned symbols are automatically evaluated (= replaced by their as-
signed value) when they are used. To refer to the unevaluated symbol,
put single quotes (') around the name, as demonstrated for the "unas-
sign" command above.
Symbols are considered to be in the complex domain by default, i.e.
they are treated as if they stand in for complex numbers. This behavior
can be changed by using the keywords real_symbols and complex_symbols
and affects all newly created symbols.
The following symbols are pre-defined constants that cannot be assigned
a value by the user:
Pi Archimedes' Constant
Catalan Catalan's Constant
Euler Euler-Mascheroni Constant
I sqrt(-1)
FAIL an object of the GiNaC "fail" class
There is also the special
Digits
symbol that controls the numeric precision of calculations with inexact
numbers. Assigning an integer value to digits will change the preci-
sion to the given number of decimal places.
WILDCARDS
The has(), find(), match() and subs() functions accept wildcards as
placeholders for expressions. These have the syntax
$number
for example $0, $1 etc.
LAST PRINTED EXPRESSIONS
ginsh provides the three special symbols
%, %% and %%%
that refer to the last, second last, and third last printed expression,
respectively. These are handy if you want to use the results of previ-
ous computations in a new expression.
OPERATORS
ginsh provides the following operators, listed in falling order of
precedence:
! postfix factorial
^ powering
+ unary plus
- unary minus
* multiplication
/ division
+ addition
- subtraction
< less than
> greater than
<= less or equal
>= greater or equal
== equal
!= not equal
= symbol assignment
All binary operators are left-associative, with the exception of ^ and
= which are right-associative. The result of the assignment operator
(=) is its right-hand side, so it's possible to assign multiple symbols
in one expression (e.g. a = b = c = 2;).
LISTS
Lists are used by the subs and lsolve functions. A list consists of an
opening curly brace ({), a (possibly empty) comma-separated sequence of
expressions, and a closing curly brace (}).
MATRICES
A matrix consists of an opening square bracket ([), a non-empty comma-
separated sequence of matrix rows, and a closing square bracket (]).
Each matrix row consists of an opening square bracket ([), a non-empty
comma-separated sequence of expressions, and a closing square bracket
(]). If the rows of a matrix are not of the same length, the width of
the matrix becomes that of the longest row and shorter rows are filled
up at the end with elements of value zero.
FUNCTIONS
A function call in ginsh has the form
name(arguments)
where arguments is a comma-separated sequence of expressions. ginsh
provides a couple of built-in functions and also "imports" all symbolic
functions defined by GiNaC and additional libraries. There is no way to
define your own functions other than linking ginsh against a library
that defines symbolic GiNaC functions.
ginsh provides Tab-completion on function names: if you type the first
part of a function name, hitting Tab will complete the name if possi-
ble. If the part you typed is not unique, hitting Tab again will dis-
play a list of matching functions. Hitting Tab twice at the prompt
will display the list of all available functions.
A list of the built-in functions follows. They nearly all work as the
respective GiNaC methods of the same name, so I will not describe them
in detail here. Please refer to the GiNaC documentation.
charpoly(matrix, symbol) - characteristic polynomial of a matrix
coeff(expression, object, number) - extracts coefficient of ob-
ject^number from a polynomial
collect(expression, object-or-list) - collects coefficients of
like powers (result in recursive form)
collect_distributed(expression, list) - collects coefficients of
like powers (result in distributed form)
collect_common_factors(expression) - collects common factors
from the terms of sums
conjugate(expression) - complex conjugation
content(expression, symbol) - content part of a polynomial
decomp_rational(expression, symbol) - decompose rational func-
tion into polynomial and proper rational function
degree(expression, object) - degree of a polynomial
denom(expression) - denominator of a rational function
determinant(matrix) - determinant of a matrix
diag(expression...) - constructs diagonal matrix
diff(expression, symbol [, number]) - partial differentiation
divide(expression, expression) - exact polynomial division
evalf(expression) - evaluates an expression to a floating point
number
evalm(expression) - evaluates sums, products and integer powers
of matrices
expand(expression) - expands an expression
factor(expression) - factorizes an expression (univariate)
find(expression, pattern) - returns a list of all occurrences of
a pattern in an expression
fsolve(expression, symbol, number, number) - numerically find
root of a real-valued function within an interval
gcd(expression, expression) - greatest common divisor
has(expression, pattern) - returns "1" if the first expression
contains the pattern as a subexpression, "0" otherwise
integer_content(expression) - integer content of a polynomial
inverse(matrix) - inverse of a matrix
is(relation) - returns "1" if the relation is true, "0" other-
wise (false or undecided)
lcm(expression, expression) - least common multiple
lcoeff(expression, object) - leading coefficient of a polynomial
ldegree(expression, object) - low degree of a polynomial
lsolve(equation-list, symbol-list) - solve system of linear
equations
map(expression, pattern) - apply function to each operand; the
function to be applied is specified as a pattern with the "$0"
wildcard standing for the operands
match(expression, pattern) - check whether expression matches a
pattern; returns a list of wildcard substitutions or "FAIL" if
there is no match
nops(expression) - number of operands in expression
normal(expression) - rational function normalization
numer(expression) - numerator of a rational function
numer_denom(expression) - numerator and denumerator of a ratio-
nal function as a list
op(expression, number) - extract operand from expression
power(expr1, expr2) - exponentiation (equivalent to writing
expr1^expr2)
prem(expression, expression, symbol) - pseudo-remainder of poly-
nomials
primpart(expression, symbol) - primitive part of a polynomial
quo(expression, expression, symbol) - quotient of polynomials
rank(matrix) - rank of a matrix
rem(expression, expression, symbol) - remainder of polynomials
resultant(expression, expression, symbol) - resultant of two
polynomials with respect to symbol s
series(expression, relation-or-symbol, order) - series expansion
series_to_poly(series) - convert a series into a polynomial by
dropping the Order() term
sprem(expression, expression, symbol) - sparse pseudo-remainder
of polynomials
sqrfree(expression [, symbol-list]) - square-free factorization
of a polynomial
sqrfree_parfrac(expression, symbol) - square-free partial frac-
tion decomposition of rational function
sqrt(expression) - square root
subs(expression, relation-or-list)
subs(expression, look-for-list, replace-by-list) - substitute
subexpressions (you may use wildcards)
tcoeff(expression, object) - trailing coefficient of a polyno-
mial
time(expression) - returns the time in seconds needed to evalu-
ate the given expression
trace(matrix) - trace of a matrix
transpose(matrix) - transpose of a matrix
unassign('symbol') - unassign an assigned symbol (mind the
quotes, please!)
unit(expression, symbol) - unit part of a polynomial
SPECIAL COMMANDS
To exit ginsh, enter
quit
or
exit
ginsh can display a (short) help for a given topic (mostly about func-
tions and operators) by entering
?topic
Typing
??
will display a list of available help topics.
The command
print(expression);
will print a dump of GiNaC's internal representation for the given ex-
pression. This is useful for debugging and for learning about GiNaC
internals.
The command
print_latex(expression);
prints a LaTeX representation of the given expression.
The command
print_csrc(expression);
prints the given expression in a way that can be used in a C or C++
program.
The command
iprint(expression);
prints the given expression (which must evaluate to an integer) in dec-
imal, octal, and hexadecimal representations.
Finally, the shell escape
! [command [arguments]]
passes the given command and optionally arguments to the shell for exe-
cution. With this method, you can execute shell commands from within
ginsh without having to quit.
EXAMPLES
> a = x^2-x-2;
-2-x+x^2
> b = (x+1)^2;
(x+1)^2
> s = a/b;
(x+1)^(-2)*(-2-x+x^2)
> diff(s, x);
(2*x-1)*(x+1)^(-2)-2*(x+1)^(-3)*(-x+x^2-2)
> normal(s);
(x-2)*(x+1)^(-1)
> x = 3^50;
717897987691852588770249
> s;
717897987691852588770247/717897987691852588770250
> Digits = 40;
40
> evalf(s);
0.999999999999999999999995821133292704384960990679
> unassign('x');
x
> s;
(x+1)^(-2)*(-x+x^2-2)
> series(sin(x),x==0,6);
1*x+(-1/6)*x^3+1/120*x^5+Order(x^6)
> lsolve({3*x+5*y == 7}, {x, y});
{x==-5/3*y+7/3,y==y}
> lsolve({3*x+5*y == 7, -2*x+10*y == -5}, {x, y});
{x==19/8,y==-1/40}
> M = [ [a, b], [c, d] ];
[[-x+x^2-2,(x+1)^2],[c,d]]
> determinant(M);
-2*d-2*x*c-x^2*c-x*d+x^2*d-c
> collect(%, x);
(-d-2*c)*x+(d-c)*x^2-2*d-c
> solve quantum field theory;
parse error at quantum
> quit
DIAGNOSTICS
parse error at foo
You entered something which ginsh was unable to parse. Please
check the syntax of your input and try again.
argument num to function must be a type
The argument number num to the given function must be of a cer-
tain type (e.g. a symbol, or a list). The first argument has
number 0, the second argument number 1, etc.
AUTHOR
The GiNaC maintainers <https://www.ginac.de/>.
SEE ALSO
GiNaC Tutorial - An open framework for symbolic computation within the
C++ programming language
CLN - A Class Library for Numbers, Bruno Haible
COPYRIGHT
Copyright © 1999-2023 Johannes Gutenberg Universität Mainz, Germany
This program is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 2 of the License, or (at your
option) any later version.
This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MER-
CHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
You should have received a copy of the GNU General Public License along
with this program; if not, write to the Free Software Foundation, 51
Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
GiNaC 1.8.6 January, 2000 ginsh(1)
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