ginsh - Online in the Cloud

PROGRAM:

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 computer 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
functions (e.g. sin or normal). Every input expression must be terminated with either a
semicolon (;) or a colon (:). If terminated with a semicolon, ginsh will evaluate the
expression and print the result to stdout. If terminated with a colon, ginsh will only
evaluate the expression but not print the result. It is possible to enter multiple
expressions 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 arbitrary precision
integers and rationals as well as floating point numbers in standard or scientific
notation (e.g. 1.2E6). The general rule is that if a number contains a decimal point
(.), it is an (inexact) floating point number; otherwise it is an (exact) integer or
rational. Integers can be specified in binary, octal, hexadecimal or arbitrary (2-36)
base by prefixing them with #b, #o, #x, or #nR , respectively.

SYMBOLS
Symbols are made up of a string of alphanumeric characters and the underscore (_), 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 assigned value) when
they are used. To refer to the unevaluated symbol, put single quotes (') around the name,
as demonstrated for the "unassign" 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 precision 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 previous 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 possible. If the part you typed is not unique,
hitting Tab again will display 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 object^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 function 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
eval(expression [, level]) - evaluates an expression, replacing symbols by their
assigned value
evalf(expression [, level]) - 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" otherwise (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 [, level]) - rational function normalization
numer(expression) - numerator of a rational function
numer_denom(expression) - numerator and denumerator of a rational 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 polynomials
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
sprem(expression, expression, symbol) - sparse pseudo-remainder of polynomials
sqrfree(expression [, symbol-list]) - square-free factorization of a polynomial
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 polynomial
time(expression) - returns the time in seconds needed to evaluate 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 functions 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 expression. 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 decimal, octal, and
hexadecimal representations.

Finally, the shell escape
! [command [arguments]]
passes the given command and optionally arguments to the shell for execution. 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 certain type (e.g. a
symbol, or a list). The first argument has number 0, the second argument number 1,
etc.

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