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5.1 Numbers | ||
5.2 Strings | ||
5.3 Constants | ||
5.4 Lists | ||
5.5 Arrays | ||
5.6 Structures |
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5.1.1 Introduction to Numbers | ||
5.1.2 Functions and Variables for Numbers |
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A complex expression is specified in Maxima by adding the real part of the
expression to %i
times the imaginary part. Thus the roots of the
equation x^2 - 4*x + 13 = 0
are 2 + 3*%i
and 2 - 3*%i
.
Note that simplification of products of complex expressions can be effected by
expanding the product. Simplification of quotients, roots, and other functions
of complex expressions can usually be accomplished by using the realpart
,
imagpart
, rectform
, polarform
, abs
, carg
functions.
Categories: Complex variables
Maxima has two types of floating point number. The first is just
called a "float" (but will be called a "machine float" for the
rest of this section to avoid ambiguity). This is stored in the
underlying lisp's DOUBLE-FLOAT type which will almost certainly be
IEEE 754 double precision floating point. To type a literal floating
point number, just type its decimal expansion (for example,
0.01
) or type it with an explicit exponent (such as 1e-2
or 0.1e-1
).
The second type of floating point number in Maxima is called a
"bigfloat". Bigfloats are stored as a mantissa and exponent in the
same way as machine floats but the exponent is an arbitrary precision
integer, so they can represent arbitrarily large or small numbers. The
user can also customise the precision of bigfloat arithmetic (which
corresponds to choosing the range of the mantissa). See fpprec
for more information. To type a literal bigfloat, use the exponent
notation as above but with the character b
in place of
e
. The example of 0.01
from above could be entered as a
bigfloat with 1b-2
or 0.001b0
.
Calculations using machine floats can be significantly faster than
using bigfloats since modern computer processors have dedicated
hardware for them. This is particularly noticeable with compiled
Maxima code. However, machine floats suffer from the problem of
overflow, where a number can become too large for its exponent to be
represented in the bits available. In interpreted code, the default
behaviour is that a calculation that would cause a floating point
overflow instead generates a bigfloat number. To configure this, see
the promote_float_to_bigfloat
variable.
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Converts all numbers and functions of numbers in expr to bigfloat numbers.
The number of significant digits in the resulting bigfloats is specified by the
global variable fpprec
.
When float2bf
is false
a warning message is printed when
a floating point number is converted into a bigfloat number (since
this may lead to loss of precision).
Categories: Numerical evaluation
Returns true
if expr is a bigfloat number, otherwise false
.
Categories: Numerical evaluation · Predicate functions
Default value: false
bftorat
controls the conversion of bfloats to rational numbers. When
bftorat
is false
, ratepsilon
will be used to control the
conversion (this results in relatively small rational numbers). When
bftorat
is true
, the rational number generated will accurately
represent the bfloat.
Note: bftorat
has no effect on the transformation to rational numbers
with the function rationalize
.
Example:
(%i1) ratepsilon:1e-4; (%o1) 1.e-4 (%i2) rat(bfloat(11111/111111)), bftorat:false; `rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1 1 (%o2)/R/ -- 10 (%i3) rat(bfloat(11111/111111)), bftorat:true; `rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2 11111 (%o3)/R/ ------ 111111
Categories: Numerical evaluation
Default value: true
bftrunc
causes trailing zeroes in non-zero bigfloat numbers not to be
displayed. Thus, if bftrunc
is false
, bfloat (1)
displays as 1.000000000000000B0
. Otherwise, this is displayed as
1.0B0
.
Categories: Numerical evaluation
Returns true
if expr is an even integer.
false
is returned in all other cases.
Categories: Predicate functions
Converts integers, rational numbers and bigfloats in expr to floating
point numbers. It is also an evflag
, float
causes
non-integral rational numbers and bigfloat numbers to be converted to floating
point.
Categories: Numerical evaluation · Evaluation flags
Default value: true
When float2bf
is false
, a warning message is printed when
a floating point number is converted into a bigfloat number (since
this may lead to loss of precision).
Categories: Numerical evaluation
Returns true
if expr is a floating point number, otherwise
false
.
Categories: Numerical evaluation · Predicate functions
Default value: 16
fpprec
is the number of significant digits for arithmetic on bigfloat
numbers. fpprec
does not affect computations on ordinary floating point
numbers.
See also bfloat
and fpprintprec
.
Categories: Numerical evaluation
Default value: 0
fpprintprec
is the number of digits to print when printing an ordinary
float or bigfloat number.
For ordinary floating point numbers,
when fpprintprec
has a value between 2 and 16 (inclusive),
the number of digits printed is equal to fpprintprec
.
Otherwise, fpprintprec
is 0, or greater than 16,
and the number of digits printed is 16.
For bigfloat numbers,
when fpprintprec
has a value between 2 and fpprec
(inclusive),
the number of digits printed is equal to fpprintprec
.
Otherwise, fpprintprec
is 0, or greater than fpprec
,
and the number of digits printed is equal to fpprec
.
For both ordinary floats and bigfloats,
trailing zero digits are suppressed.
The actual number of digits printed is less than fpprintprec
if there are trailing zero digits.
fpprintprec
cannot be 1.
Categories: Numerical evaluation · Display flags and variables
Returns true
if expr is a literal numeric integer, otherwise
false
.
integerp
returns false
if its argument is a symbol,
even if the argument is declared integer.
Examples:
(%i1) integerp (0); (%o1) true (%i2) integerp (1); (%o2) true (%i3) integerp (-17); (%o3) true (%i4) integerp (0.0); (%o4) false (%i5) integerp (1.0); (%o5) false (%i6) integerp (%pi); (%o6) false (%i7) integerp (n); (%o7) false (%i8) declare (n, integer); (%o8) done (%i9) integerp (n); (%o9) false
Categories: Predicate functions
Default value: false
m1pbranch
is the principal branch for -1
to a power.
Quantities such as (-1)^(1/3)
(that is, an "odd" rational exponent) and
(-1)^(1/4)
(that is, an "even" rational exponent) are handled as follows:
domain:real (-1)^(1/3): -1 (-1)^(1/4): (-1)^(1/4) domain:complex m1pbranch:false m1pbranch:true (-1)^(1/3) 1/2+%i*sqrt(3)/2 (-1)^(1/4) sqrt(2)/2+%i*sqrt(2)/2
Categories: Expressions · Global flags
Return true
if and only if n >= 0
and n is an integer.
Categories: Package linearalgebra · Predicate functions
Returns true
if expr is a literal integer, rational number,
floating point number, or bigfloat, otherwise false
.
numberp
returns false
if its argument is a symbol, even if the
argument is a symbolic number such as %pi
or %i
, or declared to be
even
, odd
, integer
, rational
, irrational
,
real
, imaginary
, or complex
.
Examples:
(%i1) numberp (42); (%o1) true (%i2) numberp (-13/19); (%o2) true (%i3) numberp (3.14159); (%o3) true (%i4) numberp (-1729b-4); (%o4) true (%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]); (%o5) [false, false, false, false, false, false] (%i6) declare (a, even, b, odd, c, integer, d, rational, e, irrational, f, real, g, imaginary, h, complex); (%o6) done (%i7) map (numberp, [a, b, c, d, e, f, g, h]); (%o7) [false, false, false, false, false, false, false, false]
Categories: Predicate functions
numer
causes some mathematical functions (including exponentiation)
with numerical arguments to be evaluated in floating point. It causes
variables in expr
which have been given numerals to be replaced by
their values. It also sets the float
switch on.
See also %enumer
.
Examples:
(%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))]; 1 (%o1) [sqrt(2), sin(1), -----------] sqrt(3) + 1 (%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer; (%o2) [1.414213562373095, .8414709848078965, .3660254037844387]
Categories: Numerical evaluation · Evaluation flags
Default value: false
The option variable numer_pbranch
controls the numerical evaluation of
the power of a negative integer, rational, or floating point number. When
numer_pbranch
is true
and the exponent is a floating point number
or the option variable numer
is true
too, Maxima evaluates
the numerical result using the principal branch. Otherwise a simplified, but
not an evaluated result is returned.
Examples:
(%i1) (-2)^0.75; (%o1) (-2)^0.75 (%i2) (-2)^0.75,numer_pbranch:true; (%o2) 1.189207115002721*%i-1.189207115002721 (%i3) (-2)^(3/4); (%o3) (-1)^(3/4)*2^(3/4) (%i4) (-2)^(3/4),numer; (%o4) 1.681792830507429*(-1)^0.75 (%i5) (-2)^(3/4),numer,numer_pbranch:true; (%o5) 1.189207115002721*%i-1.189207115002721
Categories: Numerical evaluation
Declares the variables x_1
, …, x_n to have
numeric values equal to expr_1
, …, expr_n
.
The numeric value is evaluated and substituted for the variable
in any expressions in which the variable occurs if the numer
flag is
true
. See also ev
.
The expressions expr_1
, …, expr_n
can be any expressions,
not necessarily numeric.
Categories: Declarations and inferences · Numerical evaluation
is true
if expr is an odd integer.
false
is returned in all other cases.
Categories: Predicate functions
Default value: true
When promote_float_to_bigfloat
is true, the result of any
floating point calculation that would normally cause a floating point
overflow is replaced by a bigfloat number that represents the
result. Note that this automatic promotion only happens in interpreted
code: compiled code is not affected.
This automatic conversion is often convenient, but can be unhelpful in
some cases. For example, it can actually cause a loss of precision if
fpprec
is currently smaller than the precision in a floating
point number. To disable this behaviour, set
promote_float_to_bigfloat
to false.
Categories: Numerical evaluation
Default value: 2.0e-15
ratepsilon
is the tolerance used in the conversion
of floating point numbers to rational numbers, when the option variable
bftorat
has the value false
. See bftorat
for an example.
Categories: Numerical evaluation · Rational expressions
Convert all double floats and big floats in the Maxima expression expr to
their exact rational equivalents. If you are not familiar with the binary
representation of floating point numbers, you might be surprised that
rationalize (0.1)
does not equal 1/10. This behavior isn't special to
Maxima - the number 1/10 has a repeating, not a terminating, binary
representation.
(%i1) rationalize (0.5); 1 (%o1) - 2 (%i2) rationalize (0.1); 1 (%o2) -- 10 (%i3) fpprec : 5$ (%i4) rationalize (0.1b0); 209715 (%o4) ------- 2097152 (%i5) fpprec : 20$ (%i6) rationalize (0.1b0); 236118324143482260685 (%o6) ---------------------- 2361183241434822606848 (%i7) rationalize (sin (0.1*x + 5.6)); x 28 (%o7) sin(-- + --) 10 5
Categories: Numerical evaluation
Returns true
if expr is a literal integer or ratio of literal
integers, otherwise false
.
Categories: Predicate functions · Rational expressions
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5.2.1 Introduction to Strings | ||
5.2.2 Functions and Variables for Strings |
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Strings (quoted character sequences) are enclosed in double quote marks "
for input, and displayed with or without the quote marks, depending on the
global variable stringdisp
.
Strings may contain any characters, including embedded tab, newline, and
carriage return characters. The sequence \"
is recognized as a literal
double quote, and \\
as a literal backslash. When backslash appears at
the end of a line, the backslash and the line termination (either newline or
carriage return and newline) are ignored, so that the string continues with the
next line. No other special combinations of backslash with another character
are recognized; when backslash appears before any character other than "
,
\
, or a line termination, the backslash is ignored. There is no way to
represent a special character (such as tab, newline, or carriage return)
except by embedding the literal character in the string.
There is no character type in Maxima; a single character is represented as a one-character string.
The stringproc
add-on package contains many functions for working with
strings.
Examples:
(%i1) s_1 : "This is a string."; (%o1) This is a string. (%i2) s_2 : "Embedded \"double quotes\" and backslash \\ characters."; (%o2) Embedded "double quotes" and backslash \ characters. (%i3) s_3 : "Embedded line termination in this string."; (%o3) Embedded line termination in this string. (%i4) s_4 : "Ignore the \ line termination \ characters in \ this string."; (%o4) Ignore the line termination characters in this string. (%i5) stringdisp : false; (%o5) false (%i6) s_1; (%o6) This is a string. (%i7) stringdisp : true; (%o7) true (%i8) s_1; (%o8) "This is a string."
Categories: Syntax
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Concatenates its arguments. The arguments must evaluate to atoms. The return value is a symbol if the first argument is a symbol and a string otherwise.
concat
evaluates its arguments. The single quote '
prevents
evaluation.
(%i1) y: 7$ (%i2) z: 88$ (%i3) concat (y, z/2); (%o3) 744 (%i4) concat ('y, z/2); (%o4) y44
A symbol constructed by concat
may be assigned a value and appear in
expressions. The ::
(double colon) assignment operator evaluates its
left-hand side.
(%i5) a: concat ('y, z/2); (%o5) y44 (%i6) a:: 123; (%o6) 123 (%i7) y44; (%o7) 123 (%i8) b^a; y44 (%o8) b (%i9) %, numer; 123 (%o9) b
Note that although concat (1, 2)
looks like a number, it is a string.
(%i10) concat (1, 2) + 3; (%o10) 12 + 3
Categories: Expressions · Strings
Concatenates its arguments into a string. Unlike concat
, the
arguments do not need to be atoms.
(%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3)); (%o1) xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
Categories: Expressions · Strings
Converts expr
to Maxima's linear notation just as if it had been typed
in.
The return value of string
is a string, and thus it cannot be used in a
computation.
Categories: Strings
Default value: false
When stringdisp
is true
, strings are displayed enclosed in double
quote marks. Otherwise, quote marks are not displayed.
stringdisp
is always true
when displaying a function definition.
Examples:
(%i1) stringdisp: false$ (%i2) "This is an example string."; (%o2) This is an example string. (%i3) foo () := print ("This is a string in a function definition."); (%o3) foo() := print("This is a string in a function definition.") (%i4) stringdisp: true$ (%i5) "This is an example string."; (%o5) "This is an example string."
Categories: Display flags and variables
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5.3.1 Functions and Variables for Constants |
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%e
represents the base of the natural logarithm, also known as Euler's
number. The numeric value of %e
is the double-precision floating-point
value 2.718281828459045d0.
Categories: Constants
%i
represents the imaginary unit, sqrt(- 1).
Categories: Constants
false
represents the Boolean constant of the same name.
Maxima implements false
by the value NIL
in Lisp.
Categories: Constants
The Euler-Mascheroni constant, 0.5772156649015329 ....
Categories: Constants
ind
represents a bounded, indefinite result.
See also limit
.
Example:
(%i1) limit (sin(1/x), x, 0); (%o1) ind
Categories: Constants
inf
represents real positive infinity.
Categories: Constants
infinity
represents complex infinity.
Categories: Constants
minf
represents real minus (i.e., negative) infinity.
Categories: Constants
%phi
represents the so-called golden mean, (1 + sqrt(5))/2.
The numeric value of %phi
is the double-precision floating-point value
1.618033988749895d0.
fibtophi
expresses Fibonacci numbers fib(n)
in terms of
%phi
.
By default, Maxima does not know the algebraic properties of %phi
.
After evaluating tellrat(%phi^2 - %phi - 1)
and algebraic: true
,
ratsimp
can simplify some expressions containing %phi
.
Examples:
fibtophi
expresses Fibonacci numbers fib(n)
in terms of %phi
.
(%i1) fibtophi (fib (n)); n n %phi - (1 - %phi) (%o1) ------------------- 2 %phi - 1 (%i2) fib (n-1) + fib (n) - fib (n+1); (%o2) - fib(n + 1) + fib(n) + fib(n - 1) (%i3) fibtophi (%); n + 1 n + 1 n n %phi - (1 - %phi) %phi - (1 - %phi) (%o3) - --------------------------- + ------------------- 2 %phi - 1 2 %phi - 1 n - 1 n - 1 %phi - (1 - %phi) + --------------------------- 2 %phi - 1 (%i4) ratsimp (%); (%o4) 0
By default, Maxima does not know the algebraic properties of %phi
.
After evaluating tellrat (%phi^2 - %phi - 1)
and algebraic: true
,
ratsimp
can simplify some expressions containing %phi
.
(%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1)); 2 2 (%o1) %phi A - %phi A - A + %phi - %phi - 1 (%i2) ratsimp (e); 2 2 (%o2) (%phi - %phi - 1) A + %phi - %phi - 1 (%i3) tellrat (%phi^2 - %phi - 1); 2 (%o3) [%phi - %phi - 1] (%i4) algebraic : true; (%o4) true (%i5) ratsimp (e); (%o5) 0
Categories: Constants
%pi
represents the ratio of the perimeter of a circle to its diameter.
The numeric value of %pi
is the double-precision floating-point value
3.141592653589793d0.
Categories: Constants
true
represents the Boolean constant of the same name.
Maxima implements true
by the value T
in Lisp.
Categories: Constants
und
represents an undefined result.
See also limit
.
Example:
(%i1) limit (x*sin(x), x, inf); (%o1) und
Categories: Constants
zeroa
represents an infinitesimal above zero. zeroa
can be used
in expressions. limit
simplifies expressions which contain
infinitesimals.
Example:
limit
simplifies expressions which contain infinitesimals:
(%i1) limit(zeroa); (%o1) 0 (%i2) limit(x+zeroa); (%o2) x
Categories: Constants
zerob
represents an infinitesimal below zero. zerob
can be used
in expressions. limit
simplifies expressions which contain
infinitesimals.
Categories: Constants
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5.4.1 Introduction to Lists | ||
5.4.2 Functions and Variables for Lists |
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Lists are the basic building block for Maxima and Lisp. All data types other than arrays, hash tables, numbers are represented as Lisp lists, These Lisp lists have the form
((MPLUS) $A 2)
to indicate an expression a+2
. At Maxima level one would see
the infix notation a+2
. Maxima also has lists which are printed
as
[1, 2, 7, x+y]
for a list with 4 elements. Internally this corresponds to a Lisp list of the form
((MLIST) 1 2 7 ((MPLUS) $X $Y ))
The flag which denotes the type field of the Maxima expression is a list itself, since after it has been through the simplifier the list would become
((MLIST SIMP) 1 2 7 ((MPLUS SIMP) $X $Y))
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[
and ]
mark the beginning and end, respectively, of a list.
[
and ]
also enclose the subscripts of
a list, array, hash array, or array function.
Examples:
(%i1) x: [a, b, c]; (%o1) [a, b, c] (%i2) x[3]; (%o2) c (%i3) array (y, fixnum, 3); (%o3) y (%i4) y[2]: %pi; (%o4) %pi (%i5) y[2]; (%o5) %pi (%i6) z['foo]: 'bar; (%o6) bar (%i7) z['foo]; (%o7) bar (%i8) g[k] := 1/(k^2+1); 1 (%o8) g := ------ k 2 k + 1 (%i9) g[10]; 1 (%o9) --- 101
Returns a single list of the elements of list_1 followed
by the elements of list_2, … append
also works on
general expressions, e.g. append (f(a,b), f(c,d,e));
yields
f(a,b,c,d,e)
.
Do example(append);
for an example.
Categories: Lists · Expressions
This function searches for key in the left hand side of the input list. The list argument should be a list, each of whose elements is an expression with exactly two parts. Most usually, the elements of list are themselves lists, each with two elements.
The assoc
function iterates along list, checking the
first part of each element for equality with key. If an element
is found where the comparison is true, assoc
returns the second
part of that element. If there is no such element in the list,
assoc
returns either false
or default, if given.
For example, in the expression assoc (y, [[x,1], [y,2],
[z,3]])
, the assoc
function searches for x
in the left
hand side of the list [[y,1],[x,2]]
and finds it at the second
term, returning 2
. In assoc (z, [[x,1], [z,2], [z,3]])
,
the search stops at the first term starting with z
and returns
2
. In assoc(x, [[y,1]])
, there is no matching element,
so assoc
returns false
.
(%i1) assoc(y, [[x, 1], [y, 2], [z, 3]]) (%o1) 2 (%i2) assoc(z, [[x, 1], [z, 2], [z, 3]]) (%o2) 2 (%i3) assoc(x, [[y, 1]]) (%o3) false
Categories: Lists · Expressions
cons (expr, list)
returns a new list constructed of the element
expr as its first element, followed by the elements of list. This is
analogous to the Lisp language construction operation "cons".
The Maxima function cons
can also be used where the second argument is other
than a list and this might be useful. In this case, cons (expr_1, expr_2)
returns an expression with same operator as expr_2 but with argument cons(expr_1, args(expr_2))
.
Examples:
(%i1) cons(a,[b,c,d]); (%o1) [a, b, c, d] (%i2) cons(a,f(b,c,d)); (%o2) f(a, b, c, d)
In general, cons
applied to a nonlist doesn't make sense. For instance, cons(a,b^c)
results in an illegal expression, since '^' cannot take three arguments.
When inflag
is true, cons
operates on the internal structure of an expression, otherwise
cons
operates on the displayed form. Especially when inflag
is true, cons
applied
to a nonlist sometimes gives a surprising result; for example
(%i1) cons(a,-a), inflag : true; 2 (%o1) - a (%i2) cons(a,-a), inflag : false; (%o2) 0
Categories: Lists · Expressions
Returns a copy of the list list.
Categories: Lists
Create a list by evaluating form with x_1 bound to each element of list_1, and for each such binding bind x_2 to each element of list_2, … The number of elements in the result will be the product of the number of elements in each list. Each variable x_i must actually be a symbol - it will not be evaluated. The list arguments will be evaluated once at the beginning of the iteration.
(%i1) create_list (x^i, i, [1, 3, 7]); 3 7 (%o1) [x, x , x ]
With a double iteration:
(%i1) create_list ([i, j], i, [a, b], j, [e, f, h]); (%o1) [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]
Instead of list_i two args may be supplied each of which should evaluate to a number. These will be the inclusive lower and upper bounds for the iteration.
(%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i); (%o1) [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]
Note that the limits or list for the j
variable can
depend on the current value of i
.
Categories: Lists
delete(expr_1, expr_2)
removes from expr_2 any arguments of its top-level operator
which are the same (as determined by "=") as expr_1.
Note that "=" tests for formal equality, not equivalence.
Note also that arguments of subexpressions are not affected.
expr_1 may be an atom or a non-atomic expression.
expr_2 may be any non-atomic expression.
delete
returns a new expression;
it does not modify expr_2.
delete(expr_1, expr_2, n)
removes from expr_2 the first n arguments of the top-level operator
which are the same as expr_1.
If there are fewer than n such arguments,
then all such arguments are removed.
Examples:
Removing elements from a list.
(%i1) delete (y, [w, x, y, z, z, y, x, w]); (%o1) [w, x, z, z, x, w]
Removing terms from a sum.
(%i1) delete (sin(x), x + sin(x) + y); (%o1) y + x
Removing factors from a product.
(%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z)); (%o1) (u - w) (u - y) (u - z)
Removing arguments from an arbitrary expression.
(%i1) delete (a, foo (a, b, c, d, a)); (%o1) foo(b, c, d)
Limit the number of removed arguments.
(%i1) delete (a, foo (a, b, a, c, d, a), 2); (%o1) foo(b, c, d, a)
Whether arguments are the same as expr_1 is determined by "=".
Arguments which are equal
but not "=" are not removed.
(%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))]; rat: replaced 0.0 by 0/1 = 0.0 `rat' replaced 0.0B0 by 0/1 = 0.0B0 (%o1) [true, true, true] (%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)]; (%o2) [true, false, false] (%i3) delete (0, [0, 0.0, 0b0]); (%o3) [0.0, 0.0b0] (%i4) is (equal ((x + y)*(x - y), x^2 - y^2)); (%o4) true (%i5) is ((x + y)*(x - y) = x^2 - y^2); (%o5) false (%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]); 2 2 (%o6) [x - y ]
Categories: Lists · Expressions
Returns the 8'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
endcons (expr, list)
returns a new list constructed of the elements of
list followed by expr. The Maxima function endcons
can also be used where
the second argument is other than a list and this might be useful. In this case,
endcons (expr_1, expr_2)
returns an expression with same operator as
expr_2 but with argument endcons(expr_1, args(expr_2))
. Examples:
(%i1) endcons(a,[b,c,d]); (%o1) [b, c, d, a] (%i2) endcons(a,f(b,c,d)); (%o2) f(b, c, d, a)
In general, endcons
applied to a nonlist doesn't make sense. For instance, endcons(a,b^c)
results in an illegal expression, since '^' cannot take three arguments.
When inflag
is true, endcons
operates on the internal structure of an expression, otherwise
endcons
operates on the displayed form. Especially when inflag
is true, endcons
applied
to a nonlist sometimes gives a surprising result; for example
(%i1) endcons(a,-a),inflag : true; 2 (%o1) - a (%i2) endcons(a,-a),inflag : false; (%o2) 0
Categories: Lists · Expressions
Returns the 5'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
Returns the first part of expr which may result in the first element of a
list, the first row of a matrix, the first term of a sum, etc. Note that
first
and its related functions, rest
and last
, work
on the form of expr which is displayed not the form which is typed on
input. If the variable inflag
is set to true
however, these
functions will look at the internal form of expr. Note that the
simplifier re-orders expressions. Thus first(x+y)
will be x
if
inflag
is true
and y
if inflag
is false
(first(y+x)
gives the same results). The functions second
…
tenth
yield the second through the tenth part of their input argument.
Categories: Lists · Expressions
Returns the 4'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
Creates a new list containing the elements of lists l and m,
interspersed. The result has elements [l[1], m[1],
l[2], m[2], ...]
. The lists l and m may contain any
type of elements.
If the lists are different lengths, join
ignores elements of the longer
list.
Maxima complains if l or m is not a list.
Examples:
(%i1) L1: [a, sin(b), c!, d - 1]; (%o1) [a, sin(b), c!, d - 1] (%i2) join (L1, [1, 2, 3, 4]); (%o2) [a, 1, sin(b), 2, c!, 3, d - 1, 4] (%i3) join (L1, [aa, bb, cc, dd, ee, ff]); (%o3) [a, aa, sin(b), bb, c!, cc, d - 1, dd]
Categories: Lists
Returns the last part (term, row, element, etc.) of the expr.
Categories: Lists · Expressions
Returns (by default) the number of parts in the external
(displayed) form of expr. For lists this is the number of elements,
for matrices it is the number of rows, and for sums it is the number
of terms (see dispform
).
The length
command is affected by the inflag
switch. So, e.g.
length(a/(b*c));
gives 2 if inflag
is false
(Assuming
exptdispflag
is true
), but 3 if inflag
is true
(the
internal representation is essentially a*b^-1*c^-1
).
Categories: Lists · Expressions
Default value: true
If false
causes any arithmetic operations with lists to be suppressed;
when true
, list-matrix operations are contagious causing lists to be
converted to matrices yielding a result which is always a matrix. However,
list-list operations should return lists.
Categories: Lists · Global flags
Returns true
if expr is a list else false
.
Categories: Lists · Predicate functions
The first form, makelist ()
, creates an empty list. The second form,
makelist (expr)
, creates a list with expr as its single
element. makelist (expr, n)
creates a list of n
elements generated from expr.
The most general form, makelist (expr, i, i_0,
i_max, step)
, returns the list of elements obtained when
ev (expr, i=j)
is applied to the elements
j of the sequence: i_0, i_0 + step, i_0 +
2*step, ..., with |j| less than or equal to |i_max|.
The increment step can be a number (positive or negative) or an expression. If it is omitted, the default value 1 will be used. If both i_0 and step are omitted, they will both have a default value of 1.
makelist (expr, x, list)
returns a list, the
j
'th element of which is equal to
ev (expr, x=list[j])
for j
equal to 1 through
length (list)
.
Examples:
(%i1) makelist (concat (x,i), i, 6); (%o1) [x1, x2, x3, x4, x5, x6] (%i2) makelist (x=y, y, [a, b, c]); (%o2) [x = a, x = b, x = c] (%i3) makelist (x^2, x, 3, 2*%pi, 2); (%o3) [9, 25] (%i4) makelist (random(6), 4); (%o4) [2, 0, 2, 5] (%i5) flatten (makelist (makelist (i^2, 3), i, 4)); (%o5) [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16] (%i6) flatten (makelist (makelist (i^2, i, 3), 4)); (%o6) [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]
Categories: Lists
Returns true
if is(expr_1 = a)
for some element a in args(expr_2)
,
otherwise returns false
.
expr_2
is typically a list, in which case
args(expr_2) = expr_2
and is(expr_1 = a)
for some element a in expr_2
is the test.
member
does not inspect parts of the arguments of expr_2
, so it
may return false
even if expr_1
is a part of some argument of
expr_2
.
See also elementp
.
Examples:
(%i1) member (8, [8, 8.0, 8b0]); (%o1) true (%i2) member (8, [8.0, 8b0]); (%o2) false (%i3) member (b, [a, b, c]); (%o3) true (%i4) member (b, [[a, b], [b, c]]); (%o4) false (%i5) member ([b, c], [[a, b], [b, c]]); (%o5) true (%i6) F (1, 1/2, 1/4, 1/8); 1 1 1 (%o6) F(1, -, -, -) 2 4 8 (%i7) member (1/8, %); (%o7) true (%i8) member ("ab", ["aa", "ab", sin(1), a + b]); (%o8) true
Categories: Lists · Expressions · Predicate functions
Returns the 9'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
pop
removes and returns the first element from the list list. The second argument
list must be a mapatom that is bound to a nonempty list. If the argument list is
not bound to a nonempty list, Maxima signals an error. For examples, see push
.
Categories: Lists · Expressions
push
prepends the item item to the list list and returns a copy of the new list.
The second argument list must be a mapatom that is bound to a list. The first argument item
can be any Maxima symbol or expression. If the argument list is not bound to a list, Maxima
signals an error.
To remove the first item from a list, see pop
.
Examples:
(%i1) ll : []; (%o1) [] (%i2) push(x,ll); (%o2) [x] (%i3) push(x^2+y,ll); 2 (%o3) [y + x , x] (%i4) push("string",ll); 2 (%o4) [string, y + x , x] (%i5) pop(ll); (%o5) string (%i6) pop(ll); 2 (%o6) y + x (%i7) pop(ll); (%o7) x (%i8) ll; (%o8) [] (%i9)
Categories: Lists · Expressions
Returns expr with its first n elements removed if n
is positive and its last - n
elements removed if n
is negative. If n is 1 it may be omitted. The first argument
expr may be a list, matrix, or other expression. When expr
is a mapatom, rest
signals an error; when expr is an empty
list and partswitch
is false, rest
signals an error. When
expr is an empty list and partswitch
is true, rest
returns end
.
Applying rest
to expression such as f(a,b,c)
returns
f(b,c)
. In general, applying rest
to an nonlist doesn't
make sense. For example, because '^' requires two arguments,
rest(a^b)
results in an error message. The functions
args
and op
may be useful as well, since args(a^b)
returns [a,b]
and op(a^b)
returns ^.
(%i1) rest(a+b+c); (%o1) b+a (%i2) rest(a+b+c,2); (%o2) a (%i3) rest(a+b+c,-2); (%o3) c
Categories: Lists · Expressions
Reverses the order of the members of the list (not
the members themselves). reverse
also works on general expressions,
e.g. reverse(a=b);
gives b=a
.
Categories: Lists · Expressions
Returns the 2'nd item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
Returns the 7'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
Returns the 6'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
sort(L, P)
sorts a list L according to a predicate P
of two arguments
which defines a strict weak order on the elements of L.
If P(a, b)
is true
, then a
appears before b
in the result.
If neither P(a, b)
nor P(b, a)
are true
,
then a
and b
are equivalent, and appear in the result in the same order as in the input.
That is, sort
is a stable sort.
If P(a, b)
and P(b, a)
are both true
for some elements of L,
then P is not a valid sort predicate, and the result is undefined.
If P(a, b)
is something other than true
or false
, sort
signals an error.
The predicate may be specified as the name of a function
or binary infix operator, or as a lambda
expression. If specified as
the name of an operator, the name must be enclosed in double quotes.
The sorted list is returned as a new object; the argument L is not modified.
sort(L)
is equivalent to sort(L, orderlessp)
.
The default sorting order is ascending, as determined by orderlessp
. The predicate ordergreatp
sorts a list in descending order.
All Maxima atoms and expressions are comparable under orderlessp
and ordergreatp
.
Operators <
and >
order numbers, constants, and constant expressions by magnitude.
Note that orderlessp
and ordergreatp
do not order numbers, constants, and constant expressions by magnitude.
ordermagnitudep
orders numbers, constants, and constant expressions the same as <
,
and all other elements the same as orderlessp
.
Examples:
sort
sorts a list according to a predicate of two arguments
which defines a strict weak order on the elements of the list.
(%i1) sort ([1, a, b, 2, 3, c], 'orderlessp); (%o1) [1, 2, 3, a, b, c] (%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp); (%o2) [c, b, a, 3, 2, 1]
The predicate may be specified as the name of a function
or binary infix operator, or as a lambda
expression. If specified as
the name of an operator, the name must be enclosed in double quotes.
(%i1) L : [[1, x], [3, y], [4, w], [2, z]]; (%o1) [[1, x], [3, y], [4, w], [2, z]] (%i2) foo (a, b) := a[1] > b[1]; (%o2) foo(a, b) := a > b 1 1 (%i3) sort (L, 'foo); (%o3) [[4, w], [3, y], [2, z], [1, x]] (%i4) infix (">>"); (%o4) >> (%i5) a >> b := a[1] > b[1]; (%o5) a >> b := a > b 1 1 (%i6) sort (L, ">>"); (%o6) [[4, w], [3, y], [2, z], [1, x]] (%i7) sort (L, lambda ([a, b], a[1] > b[1])); (%o7) [[4, w], [3, y], [2, z], [1, x]]
sort(L)
is equivalent to sort(L, orderlessp)
.
(%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi]; (%o1) [a, 2 b, - 5, 7, %e + 1, %pi] (%i2) sort (L); (%o2) [- 5, 7, %e + 1, %pi, a, 2 b] (%i3) sort (L, 'orderlessp); (%o3) [- 5, 7, %e + 1, %pi, a, 2 b]
The default sorting order is ascending, as determined by orderlessp
. The predicate ordergreatp
sorts a list in descending order.
(%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi]; (%o1) [a, 2 b, - 5, 7, %e + 1, %pi] (%i2) sort (L); (%o2) [- 5, 7, %e + 1, %pi, a, 2 b] (%i3) sort (L, 'ordergreatp); (%o3) [2 b, a, %pi, %e + 1, 7, - 5]
All Maxima atoms and expressions are comparable under orderlessp
and ordergreatp
.
(%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a]; 5 (%o1) [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a] 2 (%i2) sort (L, orderlessp); 5 (%o2) [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)] 2 (%i3) sort (L, ordergreatp); 5 (%o3) [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17] 2
Operators <
and >
order numbers, constants, and constant expressions by magnitude.
Note that orderlessp
and ordergreatp
do not order numbers, constants, and constant expressions by magnitude.
(%i1) L : [%pi, 3, 4, %e, %gamma]; (%o1) [%pi, 3, 4, %e, %gamma] (%i2) sort (L, ">"); (%o2) [4, %pi, 3, %e, %gamma] (%i3) sort (L, ordergreatp); (%o3) [%pi, %gamma, %e, 4, 3]
ordermagnitudep
orders numbers, constants, and constant expressions the same as <
,
and all other elements the same as orderlessp
.
(%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0]; (%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0] (%i2) sort (L, ordermagnitudep); (%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i, %i + 1, 2 x] (%i3) sort (L, orderlessp); (%o3) [0, 1, 1.0, 2, 3, %e, %i, %i + 1, inf, minf, sin(1), 1.0b0, 2 x]
Categories: Lists
Returns the list of elements of list for which the predicate p
returns true
.
Example:
(%i1) L: [1, 2, 3, 4, 5, 6]; (%o1) [1, 2, 3, 4, 5, 6] (%i2) sublist (L, evenp); (%o2) [2, 4, 6]
Categories: Lists
Returns the indices of the elements x
of the list L for which
the predicate maybe(P(x))
returns true
;
this excludes unknown
as well as false
.
P may be the name of a function or a lambda expression.
L must be a literal list.
Examples:
(%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], lambda ([x], x='b)); (%o1) [2, 3, 7, 9] (%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp); (%o2) [1, 2, 3, 4, 7, 9] (%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0], identity); (%o3) [1, 4, 5] (%i4) assume (x < -1); (%o4) [x < - 1] (%i5) map (maybe, [x > 0, x < 0, x < -2]); (%o5) [false, true, unknown] (%i6) sublist_indices ([x > 0, x < 0, x < -2], identity); (%o6) [2]
Categories: Lists
Returns the unique elements of the list L.
When all the elements of L are unique,
unique
returns a shallow copy of L,
not L itself.
If L is not a list, unique
returns L.
Example:
(%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]); (%o1) [1, 2, %e, %pi, [1], b + a]
Returns the 10'th item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
Returns the 3'rd item of expression or list expr.
See first
for more details.
Categories: Lists · Expressions
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5.5.1 Functions and Variables for Arrays |
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Creates an n-dimensional array. n may be less than or equal to 5. The subscripts for the i'th dimension are the integers running from 0 to dim_i.
array (name, dim_1, ..., dim_n)
creates a general
array.
array (name, type, dim_1, ..., dim_n)
creates
an array, with elements of a specified type. type can be fixnum
for integers of limited size or flonum
for floating-point numbers.
array ([name_1, ..., name_m], dim_1, ..., dim_n)
creates m arrays, all of the same dimensions.
If the user assigns to a subscripted variable before declaring the
corresponding array, an undeclared array is created.
Undeclared arrays, otherwise known as hashed arrays (because hash
coding is done on the subscripts), are more general than declared
arrays. The user does not declare their maximum size, and they grow
dynamically by hashing as more elements are assigned values. The
subscripts of undeclared arrays need not even be numbers. However,
unless an array is rather sparse, it is probably more efficient to
declare it when possible than to leave it undeclared. The array
function can be used to transform an undeclared array into a declared
array.
Categories: Arrays
Evaluates A [i_1, ..., i_n]
,
where A is an array and i_1, …, i_n are integers.
This is reminiscent of apply
, except the first argument is an array
instead of a function.
Categories: Expressions · Arrays
Returns information about the array A. The argument A may be a declared array, an undeclared (hashed) array, an array function, or a subscripted function.
For declared arrays, arrayinfo
returns a list comprising the atom
declared
, the number of dimensions, and the size of each dimension.
The elements of the array, both bound and unbound, are returned by
listarray
.
For undeclared arrays (hashed arrays), arrayinfo
returns a list
comprising the atom hashed
, the number of subscripts,
and the subscripts of every element which has a value.
The values are returned by listarray
.
For array functions, arrayinfo
returns a list comprising the atom
hashed
, the number of subscripts,
and any subscript values for which there are stored function values.
The stored function values are returned by listarray
.
For subscripted functions, arrayinfo
returns a list comprising the atom
hashed
, the number of subscripts,
and any subscript values for which there are lambda expressions.
The lambda expressions are returned by listarray
.
See also listarray
.
Examples:
arrayinfo
and listarray
applied to a declared array.
(%i1) array (aa, 2, 3); (%o1) aa (%i2) aa [2, 3] : %pi; (%o2) %pi (%i3) aa [1, 2] : %e; (%o3) %e (%i4) arrayinfo (aa); (%o4) [declared, 2, [2, 3]] (%i5) listarray (aa); (%o5) [#####, #####, #####, #####, #####, #####, %e, #####, #####, #####, #####, %pi]
arrayinfo
and listarray
applied to an undeclared (hashed) array.
(%i1) bb [FOO] : (a + b)^2; 2 (%o1) (b + a) (%i2) bb [BAR] : (c - d)^3; 3 (%o2) (c - d) (%i3) arrayinfo (bb); (%o3) [hashed, 1, [BAR], [FOO]] (%i4) listarray (bb); 3 2 (%o4) [(c - d) , (b + a) ]
arrayinfo
and listarray
applied to an array function.
(%i1) cc [x, y] := y / x; y (%o1) cc := - x, y x (%i2) cc [u, v]; v (%o2) - u (%i3) cc [4, z]; z (%o3) - 4 (%i4) arrayinfo (cc); (%o4) [hashed, 2, [4, z], [u, v]] (%i5) listarray (cc); z v (%o5) [-, -] 4 u
arrayinfo
and listarray
applied to a subscripted function.
(%i1) dd [x] (y) := y ^ x; x (%o1) dd (y) := y x (%i2) dd [a + b]; b + a (%o2) lambda([y], y ) (%i3) dd [v - u]; v - u (%o3) lambda([y], y ) (%i4) arrayinfo (dd); (%o4) [hashed, 1, [b + a], [v - u]] (%i5) listarray (dd); b + a v - u (%o5) [lambda([y], y ), lambda([y], y )]
Categories: Arrays
Returns the expression A[i_1, ..., i_n]
.
The result is an unevaluated array reference.
arraymake
is reminiscent of funmake
, except the return value
is an unevaluated array reference instead of an unevaluated function call.
Examples:
(%i1) arraymake (A, [1]); (%o1) A 1 (%i2) arraymake (A, [k]); (%o2) A k (%i3) arraymake (A, [i, j, 3]); (%o3) A i, j, 3 (%i4) array (A, fixnum, 10); (%o4) A (%i5) fillarray (A, makelist (i^2, i, 1, 11)); (%o5) A (%i6) arraymake (A, [5]); (%o6) A 5 (%i7) ''%; (%o7) 36 (%i8) L : [a, b, c, d, e]; (%o8) [a, b, c, d, e] (%i9) arraymake ('L, [n]); (%o9) L n (%i10) ''%, n = 3; (%o10) c (%i11) A2 : make_array (fixnum, 10); (%o11) {Array: #(0 0 0 0 0 0 0 0 0 0)} (%i12) fillarray (A2, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]); (%o12) {Array: #(1 2 3 4 5 6 7 8 9 10)} (%i13) arraymake ('A2, [8]); (%o13) A2 8 (%i14) ''%; (%o14) 9
Categories: Expressions · Arrays
Default value: []
arrays
is a list of arrays that have been allocated.
These comprise arrays declared by array
, hashed arrays constructed by
implicit definition (assigning something to an array element),
and array functions defined by :=
and define
.
Arrays defined by make_array
are not included.
See also
array
, arrayapply
, arrayinfo
,
arraymake
, fillarray
, listarray
, and
rearray
.
Examples:
(%i1) array (aa, 5, 7); (%o1) aa (%i2) bb [FOO] : (a + b)^2; 2 (%o2) (b + a) (%i3) cc [x] := x/100; x (%o3) cc := --- x 100 (%i4) dd : make_array ('any, 7); (%o4) {Array: #(NIL NIL NIL NIL NIL NIL NIL)} (%i5) arrays; (%o5) [aa, bb, cc]
Categories: Arrays · Global variables
Assigns x to A[i_1, ..., i_n]
,
where A is an array and i_1, …, i_n are integers.
arraysetapply
evaluates its arguments.
Categories: Expressions · Arrays
Fills array A from B, which is a list or an array.
If a specific type was declared for A when it was created, it can only be filled with elements of that same type; it is an error if an attempt is made to copy an element of a different type.
If the dimensions of the arrays A and B are different, A is filled in row-major order. If there are not enough elements in B the last element is used to fill out the rest of A. If there are too many, the remaining ones are ignored.
fillarray
returns its first argument.
Examples:
Create an array of 9 elements and fill it from a list.
(%i1) array (a1, fixnum, 8); (%o1) a1 (%i2) listarray (a1); (%o2) [0, 0, 0, 0, 0, 0, 0, 0, 0] (%i3) fillarray (a1, [1, 2, 3, 4, 5, 6, 7, 8, 9]); (%o3) a1 (%i4) listarray (a1); (%o4) [1, 2, 3, 4, 5, 6, 7, 8, 9]
When there are too few elements to fill the array, the last element is repeated. When there are too many elements, the extra elements are ignored.
(%i1) a2 : make_array (fixnum, 8); (%o1) {Array: #(0 0 0 0 0 0 0 0)} (%i2) fillarray (a2, [1, 2, 3, 4, 5]); (%o2) {Array: #(1 2 3 4 5 5 5 5)} (%i3) fillarray (a2, [4]); (%o3) {Array: #(4 4 4 4 4 4 4 4)} (%i4) fillarray (a2, makelist (i, i, 1, 100)); (%o4) {Array: #(1 2 3 4 5 6 7 8)}
Multple-dimension arrays are filled in row-major order.
(%i1) a3 : make_array (fixnum, 2, 5); (%o1) {Array: #2A((0 0 0 0 0) (0 0 0 0 0))} (%i2) fillarray (a3, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]); (%o2) {Array: #2A((1 2 3 4 5) (6 7 8 9 10))} (%i3) a4 : make_array (fixnum, 5, 2); (%o3) {Array: #2A((0 0) (0 0) (0 0) (0 0) (0 0))} (%i4) fillarray (a4, a3); (%o4) {Array: #2A((1 2) (3 4) (5 6) (7 8) (9 10))}
Categories: Arrays
Returns a list of the elements of the array A. The argument A may be a declared array, an undeclared (hashed) array, an array function, or a subscripted function.
Elements are listed in row-major order.
That is, elements are sorted according to the first index, then according to
the second index, and so on. The sorting order of index values is the same as
the order established by orderless
.
For undeclared arrays, array functions, and subscripted functions,
the elements correspond to the index values returned by arrayinfo
.
Unbound elements of declared general arrays (that is, not fixnum
and not
flonum
) are returned as #####
.
Unbound elements of declared fixnum
or flonum
arrays
are returned as 0 or 0.0, respectively.
Unbound elements of undeclared arrays, array functions,
and subscripted functions are not returned.
Examples:
listarray
and arrayinfo
applied to a declared array.
(%i1) array (aa, 2, 3); (%o1) aa (%i2) aa [2, 3] : %pi; (%o2) %pi (%i3) aa [1, 2] : %e; (%o3) %e (%i4) listarray (aa); (%o4) [#####, #####, #####, #####, #####, #####, %e, #####, #####, #####, #####, %pi] (%i5) arrayinfo (aa); (%o5) [declared, 2, [2, 3]]
listarray
and arrayinfo
applied to an undeclared (hashed) array.
(%i1) bb [FOO] : (a + b)^2; 2 (%o1) (b + a) (%i2) bb [BAR] : (c - d)^3; 3 (%o2) (c - d) (%i3) listarray (bb); 3 2 (%o3) [(c - d) , (b + a) ] (%i4) arrayinfo (bb); (%o4) [hashed, 1, [BAR], [FOO]]
listarray
and arrayinfo
applied to an array function.
(%i1) cc [x, y] := y / x; y (%o1) cc := - x, y x (%i2) cc [u, v]; v (%o2) - u (%i3) cc [4, z]; z (%o3) - 4 (%i4) listarray (cc); z v (%o4) [-, -] 4 u (%i5) arrayinfo (cc); (%o5) [hashed, 2, [4, z], [u, v]]
listarray
and arrayinfo
applied to a subscripted function.
(%i1) dd [x] (y) := y ^ x; x (%o1) dd (y) := y x (%i2) dd [a + b]; b + a (%o2) lambda([y], y ) (%i3) dd [v - u]; v - u (%o3) lambda([y], y ) (%i4) listarray (dd); b + a v - u (%o4) [lambda([y], y ), lambda([y], y )] (%i5) arrayinfo (dd); (%o5) [hashed, 1, [b + a], [v - u]]
Categories: Arrays
Creates and returns a Lisp array. type may
be any
, flonum
, fixnum
, hashed
or
functional
.
There are n indices,
and the i'th index runs from 0 to dim_i - 1.
The advantage of make_array
over array
is that the return value
doesn't have a name, and once a pointer to it goes away, it will also go away.
For example, if y: make_array (...)
then y
points to an object
which takes up space, but after y: false
, y
no longer
points to that object, so the object can be garbage collected.
Examples:
(%i1) A1 : make_array (fixnum, 10); (%o1) {Array: #(0 0 0 0 0 0 0 0 0 0)} (%i2) A1 [8] : 1729; (%o2) 1729 (%i3) A1; (%o3) {Array: #(0 0 0 0 0 0 0 0 1729 0)} (%i4) A2 : make_array (flonum, 10); (%o4) {Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)} (%i5) A2 [2] : 2.718281828; (%o5) 2.718281828 (%i6) A2; (%o6) {Array: #(0.0 0.0 2.718281828 0.0 0.0 0.0 0.0 0.0 0.0 0.0)} (%i7) A3 : make_array (any, 10); (%o7) {Array: #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)} (%i8) A3 [4] : x - y - z; (%o8) - z - y + x (%i9) A3; (%o9) {Array: #(NIL NIL NIL NIL ((MPLUS SIMP) $X ((MTIMES SIMP)\ -1 $Y) ((MTIMES SIMP) -1 $Z)) NIL NIL NIL NIL NIL)} (%i10) A4 : make_array (fixnum, 2, 3, 5); (%o10) {Array: #3A(((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0)) ((0 0 \ 0 0 0) (0 0 0 0 0) (0 0 0 0 0)))} (%i11) fillarray (A4, makelist (i, i, 1, 2*3*5)); (%o11) {Array: #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 15)) ((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29 30)))} (%i12) A4 [0, 2, 1]; (%o12) 12
Categories: Arrays
Changes the dimensions of an array.
The new array will be filled with the elements of the old one in
row-major order. If the old array was too small,
the remaining elements are filled with
false
, 0.0
or 0
,
depending on the type of the array. The type of the array cannot be
changed.
Categories: Arrays
Removes arrays and array associated functions and frees the storage occupied. The arguments may be declared arrays, undeclared (hashed) arrays, array functions, and subscripted functions.
remarray (all)
removes all items in the global list arrays
.
It may be necessary to use this function if it is desired to redefine the values in a hashed array.
remarray
returns the list of arrays removed.
remarray
quotes its arguments.
Categories: Arrays
Evaluates the subscripted expression x[i]
.
subvar
evaluates its arguments.
arraymake (x, [i])
constructs the expression
x[i]
, but does not evaluate it.
Examples:
(%i1) x : foo $ (%i2) i : 3 $ (%i3) subvar (x, i); (%o3) foo 3 (%i4) foo : [aa, bb, cc, dd, ee]$ (%i5) subvar (x, i); (%o5) cc (%i6) arraymake (x, [i]); (%o6) foo 3 (%i7) ''%; (%o7) cc
Categories: Expressions · Arrays
Returns true
if expr is a subscripted variable, for example
a[i]
.
Categories: Predicate functions
If true
then only two types of arrays are recognized:
a:make_array(any,3,4);
then a
will have as value, an array with twelve slots, and the
indexing is zero based.
b[x+1]:y^2
(and b
is not already an array, a list, or a
matrix - if it were one of these an error would be caused since
x+1
would not be a valid subscript for an art-q array, a list or
a matrix). Its indices (also known as keys) may be any object.
It only takes one key at a time (b[x+1,u]:y
would ignore the u
).
Referencing is done by b[x+1] ==> y^2
. Of course the key may be
a list, e.g. b[[x+1,u]]:y
would be valid. This is incompatible
with the old Maxima hash arrays, but saves consing.
An advantage of storing the arrays as values of the symbol is that the
usual conventions about local variables of a function apply to arrays as
well. The Hash_table type also uses less consing and is more efficient
than the old type of Maxima hashar. To obtain consistent behaviour in
translated and compiled code set translate_fast_arrays
to be
true
.
Categories: Arrays · Global flags
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5.6.1 Introduction to Structures | ||
5.6.2 Functions and Variables for Structures |
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Maxima provides a simple data aggregate called a structure. A structure is an expression in which arguments are identified by name (the field name) and the expression as a whole is identified by its operator (the structure name). A field value can be any expression.
A structure is defined by the defstruct
function;
the global variable structures
is the list of user-defined structures.
The function new
creates instances of structures.
The @
operator refers to fields.
kill(S)
removes the structure definition S,
and kill(x@ a)
unbinds the field a of the structure instance x.
In the pretty-printing console display (with display2d
equal to true
),
structure instances are displayed with the value of each field
represented as an equation, with the field name on the left-hand side
and the value on the right-hand side.
(The equation is only a display construct; only the value is actually stored.)
In 1-dimensional display (via grind
or with display2d
equal to false
),
structure instances are displayed without the field names.
There is no way to use a field name as a function name, although a field value can be a lambda expression. Nor can the values of fields be restricted to certain types; any field can be assigned any kind of expression. There is no way to make some fields accessible or inaccessible in different contexts; all fields are always visible.
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structures
is the list of user-defined structures defined by defstruct
.
Categories: Structures · Global variables
Define a structure, which is a list of named fields a_1, …,
a_n associated with a symbol S.
An instance of a structure is just an expression which has operator S
and exactly n
arguments.
new(S)
creates a new instance of structure S.
An argument which is just a symbol a specifies the name of a field.
An argument which is an equation a = v
specifies the field name a
and its default value v.
The default value can be any expression.
defstruct
puts S on the list of user-defined structures, structures
.
kill(S)
removes S from the list of user-defined structures,
and removes the structure definition.
Examples:
(%i1) defstruct (foo (a, b, c)); (%o1) [foo(a, b, c)] (%i2) structures; (%o2) [foo(a, b, c)] (%i3) new (foo); (%o3) foo(a, b, c) (%i4) defstruct (bar (v, w, x = 123, y = %pi)); (%o4) [bar(v, w, x = 123, y = %pi)] (%i5) structures; (%o5) [foo(a, b, c), bar(v, w, x = 123, y = %pi)] (%i6) new (bar); (%o6) bar(v, w, x = 123, y = %pi) (%i7) kill (foo); (%o7) done (%i8) structures; (%o8) [bar(v, w, x = 123, y = %pi)]
Categories: Structures
new
creates new instances of structures.
new(S)
creates a new instance of structure S
in which each field is assigned its default value, if any,
or no value at all if no default was specified in the structure definition.
new(S(v_1, ..., v_n))
creates a new instance of S
in which fields are assigned the values v_1, …, v_n.
Examples:
(%i1) defstruct (foo (w, x = %e, y = 42, z)); (%o1) [foo(w, x = %e, y = 42, z)] (%i2) new (foo); (%o2) foo(w, x = %e, y = 42, z) (%i3) new (foo (1, 2, 4, 8)); (%o3) foo(w = 1, x = 2, y = 4, z = 8)
Categories: Structures
@
is the structure field access operator.
The expression x@ a
refers to the value of field a of the structure instance x.
The field name is not evaluated.
If the field a in x has not been assigned a value,
x@ a
evaluates to itself.
kill(x@ a)
removes the value of field a in x.
Examples:
(%i1) defstruct (foo (x, y, z)); (%o1) [foo(x, y, z)] (%i2) u : new (foo (123, a - b, %pi)); (%o2) foo(x = 123, y = a - b, z = %pi) (%i3) u@z; (%o3) %pi (%i4) u@z : %e; (%o4) %e (%i5) u; (%o5) foo(x = 123, y = a - b, z = %e) (%i6) kill (u@z); (%o6) done (%i7) u; (%o7) foo(x = 123, y = a - b, z) (%i8) u@z; (%o8) u@z
The field name is not evaluated.
(%i1) defstruct (bar (g, h)); (%o1) [bar(g, h)] (%i2) x : new (bar); (%o2) bar(g, h) (%i3) x@h : 42; (%o3) 42 (%i4) h : 123; (%o4) 123 (%i5) x@h; (%o5) 42 (%i6) x@h : 19; (%o6) 19 (%i7) x; (%o7) bar(g, h = 19) (%i8) h; (%o8) 123
Categories: Structures · Operators
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