5.1.1 Introduction to Numbers

Complex numbers

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.

5.1.2 Functions and Variables for Numbers

Function: bfloat (expr) ¶

bfloat replaces integers, rationals, floating point numbers, and some symbolic constants in expr with bigfloat (variable-precision floating point) numbers.

The constants %e, %gamma, %phi, and %pi are replaced by a numerical approximation. However, %e in %e^x is not replaced by a numeric value unless bfloat(x) is a number.

bfloat also causes numerical evaluation of some built-in functions, namely trigonometric functions, exponential functions, abs, and log.

The number of significant digits in the resulting bigfloats is specified by the global variable fpprec. Bigfloats already present in expr are replaced with values which have precision specified by the current value of fpprec.

When float2bf is false, a warning message is printed when a floating point number is replaced by a bigfloat number with less precision.

Examples:

bfloat replaces integers, rationals, floating point numbers, and some symbolic constants in expr with bigfloat numbers.

(%i1) bfloat([123, 17/29, 1.75]);
(%o1)        [1.23b2, 5.862068965517241b-1, 1.75b0]
(%i2) bfloat([%e, %gamma, %phi, %pi]);
(%o2) [2.718281828459045b0, 5.772156649015329b-1, 
                        1.618033988749895b0, 3.141592653589793b0]
(%i3) bfloat((f(123) + g(h(17/29)))/(x + %gamma));
         1.0b0 (g(h(5.862068965517241b-1)) + f(1.23b2))
(%o3)    ----------------------------------------------
                    x + 5.772156649015329b-1

bfloat also causes numerical evaluation of some built-in functions.

(%i1) bfloat(sin(17/29));
(%o1)                 5.532051841609784b-1
(%i2) bfloat(exp(%pi));
(%o2)                  2.314069263277927b1
(%i3) bfloat(abs(-%gamma));
(%o3)                 5.772156649015329b-1
(%i4) bfloat(log(%phi));
(%o4)                 4.812118250596035b-1

Categories: Numerical evaluation ·

Function: bfloatp (expr) ¶

Returns true if expr is a bigfloat number, otherwise false.

Categories: Numerical evaluation · Predicate functions ·

Option variable: bftorat ¶

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)                         1.0e-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 ·

Option variable: bftrunc ¶

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 ·

Function: bigfloat_bits () ¶

Returns the number of bits of precision in a bigfloat number. This value depends, of course, on the value of fpprec.

(%i1) fpprec:16;
(%o1)                                 16
(%i2) bigfloat_bits();
(%o2)                                 56
(%i3) fpprec:32;
(%o3)                                 32
(%i4) bigfloat_bits();
(%o4)                                 109

Categories: Numerical evaluation ·

Function: bigfloat_eps () ¶

Returns the smallest bigfloat value, eps, such that 1+eps is not equal to 1. The value depends on fpprec, of course.

(%i1) fpprec:16;
(%o1)                                 16
(%i2) bigfloat_eps();
(%o2)                        1.387778780781446b-17
(%i3) fpprec:32;
(%o3)                                 32
(%i4) bigfloat_eps();
(%o4)                1.5407439555097886824447823540679b-33

Categories: Numerical evaluation ·

Function: decode_float (f) ¶

decode_float takes a float f and returns a list of three values that characterizes f, which must be either a float or bfloat. The first value has the same type as f, but is a number in the range [1, 2). The second value is an exponent. The third value is a float of the same type as f and has the value of 1 if f is greater than or equal to 0; otherwise, -1.

If the returned list is [mantissa, expo, sign], then scale_float(mantissa, exp)*sign is identical to f.

(%i1) decode_float(4e0);
(%o1)                            [1.0, 2, 1.0]
(%i2) decode_float(4b0);
(%o2)                          [1.0b0, 2, 1.0b0]
(%i3) decode_float(%pi);

decode_float is only defined for floats and bfloats: %pi
 -- an error. To debug this try: debugmode(true);
(%i4) decode_float(float(%pi));
(%o4)                     [1.570796326794897, 1, 1.0]
(%i5) decode_float(1.1e-5);
(%o5)                        [1.441792, - 17, 1.0]
(%i6) %[1]*2^%[2];
(%o6)                               1.1e-5

This is a relatively simple interface to Common Lisp decode_float. However we return a signficand in the range [1,2) instead of [0.5, 1). The former matches IEEE-754. Of course, this is extended to support bfloats.

Categories: Numerical evaluation ·

Function: evenp (expr) ¶

Returns true if expr is a literal even integer, otherwise false.

evenp returns false if expr is a symbol, even if expr is declared even.

Categories: Predicate functions ·

Function: float (expr) ¶

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 ·

Option variable: float2bf ¶

Default value: true

When float2bf is false, a warning message is printed when a floating point number is replaced by a bigfloat number with less precision.

Categories: Numerical evaluation ·

Function: float_bits () ¶

Returns the number of bits of precision of a floating-point number.

Categories: Numerical evaluation ·

Function: float_eps () ¶

Returns the smallest floating-point value, eps, such that 1+eps is not equal to 1.

Categories: Numerical evaluation ·

Function: float_precision (f) ¶

Returns the number of bits of precision of a floating-point number, which can be either a float or bigfloat. This is basically the number of bits used to represent the mantissa of a floating-point number. For floats, this is 53 (for IEEE double-floats), but can be less when denormal numbers occur. For bigfloats, this is equal to fpprec, when converted from digits to bits.

Categories: Numerical evaluation ·

Function: float_sign (f) ¶

Returns the sign of f. It is +1 or -1 of the same type as f. It is an error if f is not a float or bigfloat. Note that some lisps do not support signed zeros for floating-point numbers. Bigfloats do not support signed zeroes. The examples below assume signed zeroes are supported.

(%i1) float_sign(1.0);
(%o1)                                 1.0
(%i2) float_sign(-5.0);
(%o2)                                - 1.0
(%i3) float_sign(-0.0);
(%o3)                                - 1.0
(%i4) float_sign(1b0);
(%o4)                                1.0b0
(%i5) float_sign(-5b0);
(%o5)                               - 1.0b0
(%o6) float_sign(-0b0);
(%o6)                                1.0b0
(%i7) float_sign(%pi);

float_sign is only defined for floats and bfloats: %pi
 -- an error. To debug this try: debugmode(true);

Function: floatnump (expr) ¶

Returns true if expr is a floating point number, otherwise false.

Categories: Numerical evaluation · Predicate functions ·

Option variable: fpprec ¶

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 ·

Option variable: fpprintprec ¶

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 is printed "readably": that is, it is printed with sufficient digits to exactly reconstruct the number on input.

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 ·

Function: integerp (expr) ¶

Returns true if expr is a literal numeric integer, otherwise false.

integerp returns false if expr is a symbol, even if expr 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 ·

Function: integer_decode_float (f) ¶

integer_decode_float takes a float f and returns a list of three values that characterizes f, which must be either a float or bfloat. The first value is an integer. The second value is an exponent. The third value is 1 if f is positive or zero; otherwise, -1.

If the returned list is [mantissa, expo, sign], then scale_float(fl(mantissa), expo)*sign is identical to f. Here, fl is either float or bfloat depending on whether f is a float or a bfloat.

(%i1) integer_decode_float(4.0);
(%o1)                     [4503599627370496, - 50, 1]
(%i2) integer_decode_float(4b0);
(%o2)                    [36028797018963968, - 53, 1]
(%i3) scale_float(float(%o1[1]), %o1[2]);
(%o3)                                 4.0
(%i4) scale_float(bfloat(%o2[1]), %o2[2]);
(%o4)                                4.0b0
(%i5) integer_decode_float(4);

decode_float is only defined for floats and bfloats: 4
 -- an error. To debug this try: debugmode(true);
(%i6) integer_decode_float(1e-7);
(%o6)                     [7555786372591432, - 76, 1]
(%i7) integer_decode_float(1b-7);
(%o7)                    [60446290980731459, - 79, 1]
(%i8) scale_float(float(%o6[1]), %o6[2]);
(%o8)                               1.0e-7

For lisps that support denormal numbers, we have the following results.

(%i1) integer_decode_float(least_positive_float);
(%o1)                           [1, - 1074, 1]
(%i2) integer_decode_float(100*least_positive_float);
(%o2)                          [100, - 1074, 1]
(%i3) integer_decode_float(least_positive_normalized_float);
(%o3)                    [4503599627370496, - 1074, 1]

The number of bits in the integer part decreases as the denormal number decreases. Bfloat numbers do not have denormals because the exponent is not bounded.

This is a relatively simple interface to Common Lisp integer_decode_float. However, the integer part can vary depending on the Lisp implementation; we return the same value, independent of the Lisp implementation. Of course, this is extended to support bfloats.

Categories: Numerical evaluation ·

Function: is_power_of_two (n) ¶

is_power_to_two returns true if n is a power of two and false otherwise. n may be an integer, a rational, a float, or a big float.

Some examples:

(%i1) is_power_of_two(0);
(%o1)                                false
(%i2) is_power_of_two(4);
(%o2)                                true
(%i3) is_power_of_two(355/113);
(%o3)                                false
(%i4) is_power_of_two(1/32);
(%o4)                                true
(%i5) is_power_of_two(1048576);
(%o5)                                true
(%i6) is_power_of_two(1048575);
(%o6)                                false
(%i7) is_power_of_two(0.0);
(%o7)                                false
(%i8) is_power_of_two(1048576.0);
(%o8)                                true
(%i9) is_power_of_two(1048575.0);
(%o9)                                false
(%i10) is_power_of_two(1/256.0);
(%o10)                               true
(%i11) is_power_of_two(0b0);
(%o11)                               false
(%i12) is_power_of_two(1048576b0);
(%o12)                               true
(%i13) is_power_of_two(1048575b0);
(%o13)                               false
(%i14) is_power_of_two(1/256b0);
(%o14)                               true

Categories: Predicate functions ·

Option variable: m1pbranch ¶

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 ·

Function: nonnegintegerp (n) ¶

Return true if and only if n >= 0 and n is an integer.

Categories: Predicate functions ·

Function: numberp (expr) ¶

Returns true if expr is a literal integer, rational number, floating point number, or bigfloat, otherwise false.

numberp returns false if expr is a symbol, even if expr 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 ·

Option variable: numer ¶

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, 0.8414709848078965, 0.3660254037844387]

Categories: Numerical evaluation · Evaluation flags ·

Option variable: numer_pbranch ¶

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;
                                 0.75
(%o1)                       (- 2)

(%i2) (-2)^0.75,numer_pbranch:true;
(%o2)       1.189207115002721 %i - 1.189207115002721

(%i3) (-2)^(3/4);
                               3/4  3/4
(%o3)                     (- 1)    2

(%i4) (-2)^(3/4),numer;
                                          0.75
(%o4)              1.681792830507429 (- 1)

(%i5) (-2)^(3/4),numer,numer_pbranch:true;
(%o5)       1.189207115002721 %i - 1.189207115002721

Categories: Numerical evaluation ·

Function: numerval (x_1, expr_1, …, var_n, expr_n) ¶

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 ·

Function: oddp (expr) ¶

Returns true if expr is a literal odd integer, otherwise false.

oddp returns false if expr is a symbol, even if expr is declared odd.

Categories: Predicate functions ·

Option variable: ratepsilon ¶

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 ·

Function: rationalize (expr) ¶

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);
                        3602879701896397
(%o2)                   -----------------
                        36028797018963968

(%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));
               3602879701896397 x   3152519739159347
(%o7)      sin(------------------ + ----------------)
               36028797018963968    562949953421312

Categories: Numerical evaluation ·

Function: ratnump (expr) ¶

Returns true if expr is a literal integer or ratio of literal integers, otherwise false.

Categories: Predicate functions · Rational expressions ·

Function: scale_float (f, n) ¶

scale_float scales the float f by the value 2^n. This is done carefully so that no round-off every occurs. If f is a float, then it is possible to underflow to 0 or overflow, depending on the value of f and n. Bigfloats cannot underflow or overflow.

(%i1) scale_float(2d0, 2);
(%o1)                                 8.0
(%i2) scale_float(2d0, -2);
(%o2)                                 0.5
(%i3) scale_float(-2d0, -10);
(%o3)                            - 0.001953125
(%i4) scale_float(1d0, -2000);
(%o4)                                 0.0
(%i5) scale_float(2b0, 2);
(%o5)                                8.0b0
(%i6) scale_float(1b0, -2000);
(%o6)                       8.709809816217217b-603
(%i7) scale_float(1, 5);

scale_float: first arg must be a float or bfloat: 1
 -- an error. To debug this try: debugmode(true);
(%i8) scale_float(1.0, n);

scale_float: second arg must be an integer: n
 -- an error. To debug this try: debugmode(true);

This is a relatively simple interface to Common Lisp scale_float. Of course, this is extended to support bfloats.

Function: unit_in_last_plase (n) ¶

unit_in_last_place returns a value that is the gap between n and the nearest other number. See, for example, Kahan, FOOTNOTE 1. unit_in_last_place supports rational numbers, floating-point numbers and bigfloat numbers. For integer, the result is always 1, and for rational numbers the result is always 0.

The examples below assume IEEE-754 arithmetic that supports denormal numbers. Some lisps like Clisp do not have denormal numbers.

(%i1) unit_in_last_place(0);
(%o1)                                  1
(%i2) unit_in_last_place(-123);
(%o2)                                  1
(%i3) unit_in_last_place(2/3);
(%o3)                                  0
(%i4) unit_in_last_place(355/113);
(%o4)                                  0
(%i5) unit_in_last_place(0b0);
(%o5)                                0.0b0
(%i6) unit_in_last_place(0.0);
(%o6)                       4.940656458412465e-324
(%i7) unit_in_last_place(1.0);
(%o7)                        1.110223024625157e-16
(%i8) unit_in_last_place(1b0);
(%o8)                        1.387778780781446b-17
(%i9) unit_in_last_place(100.0);
(%o9)                         1.4210854715202e-14
(%i10) unit_in_last_place(100b0);
(%o10)                       1.77635683940025b-15
(%i11) fpprec:32;
(%o11)                                32
(%i12) unit_in_last_place(1b0);
(%o12)               1.5407439555097886824447823540679b-33
(%i13) unit_in_last_place(100b0);
(%o13)               1.972152263052529513529321413207b-31

Categories: Numerical evaluation ·

5.2.1 Introduction to Strings

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."

5.2.2 Functions and Variables for Strings

Function: concat (arg_1, arg_2, …) ¶

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.

See also sconcat, that works on non-atoms, too, simplode, string and eval_string. For complex string conversions see also printf.

(%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 ·

Function: sconcat (arg_1, arg_2, …) ¶

Concatenates its arguments into a string. Unlike concat, the arguments do not need to be atoms.

See also concat, simplode, string and eval_string. For complex string conversions see also printf.

(%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3));
(%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3

Another purpose for sconcat is to convert arbitrary objects to strings.

(%i1) sconcat (x);
(%o1)                           x

(%i2) stringp(%);
(%o2)                         true

Categories: Expressions · Strings ·

Function: string (expr) ¶

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.

See also concat, sconcat, simplode and eval_string.

Categories: Strings ·

Option variable: stringdisp ¶

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 ·

5.3.1 Functions and Variables for Constants

Constant: %e ¶

%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. (See A&S eqn 4.1.16, A&S 4.1.17.)

Categories: Constants ·

Constant: %i ¶

%i represents the imaginary unit, \(\sqrt{-1}.\)

Categories: Constants ·

Constant: false ¶

false represents the Boolean constant of the same name. Maxima implements false by the value NIL in Lisp.

Categories: Constants ·

Constant: %gamma ¶

The Euler-Mascheroni constant, 0.5772156649015329.... It is defined by (A&S eqn 6.1.3 and DLMF 5.2.ii) $$ \gamma = \lim_{n \rightarrow \infty} \left(\sum_{k=1}^n {1\over k} - \log n\right) $$

Categories: Constants ·

Constant: ind ¶

ind represents a bounded, indefinite result.

See also limit.

Example:

(%i1) limit (sin(1/x), x, 0);
(%o1)                          ind

Categories: Constants ·

Constant: inf ¶

inf represents real positive infinity.

Categories: Constants ·

Constant: infinity ¶

infinity represents complex infinity.

Categories: Constants ·

Constant: least_negative_float ¶

The least negative floating-point number in Maxima. That is, the negative floating-point number closest to 0. It is approximately -4.94065e-324, when denormal numbers are supported. Otherwise it is the same as least_negative_normalized_float.

Categories: Constants ·

Constant: least_negative_normalized_float ¶

The least negative normalized floating-point number in Maxima. That is, the negative normalized floating-point number closest to 0. It is approximately -2.22507e-308.

Categories: Constants ·

Constant: least_positive_float ¶

The least positive floating-point number in Maxima. That is, the positive floating-point number closest to 0. It is approximately 4.94065e-324, when denormal numbers are supported. Otherwise it is the same as least_positive_normalized_float.

Categories: Constants ·

Constant: least_positive_normalized_float ¶

The least positive normalized floating-point number in Maxima. That is, the positive normalized floating-point number closest to 0. It is approximately 2.22507e-308.

Categories: Constants ·

Constant: minf ¶

minf represents real minus (i.e., negative) infinity.

Categories: Constants ·

Constant: most_negative_float ¶

The most negative floating-point number in Maxima. It is approximately -1.79769e+308.

Categories: Constants ·

Constant: most_positive_float ¶

The most positive floating-point number in Maxima. It is approximately 1.797693e+308.

Categories: Constants ·

Constant: %phi ¶

%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 ·

Constant: %pi ¶

%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 ·

Constant: true ¶

true represents the Boolean constant of the same name. Maxima implements true by the value T in Lisp.

Categories: Constants ·

Constant: und ¶

und represents an undefined result.

See also limit.

Example:

(%i1) limit (x*sin(x), x, inf);
(%o1)                          und

Categories: Constants ·

Constant: zeroa ¶

zeroa represents an infinitesimal above zero. zeroa can be used in expressions. limit simplifies expressions which contain infinitesimals.

See also zerob and limit.

Example:

limit simplifies expressions which contain infinitesimals:

(%i1) limit(zeroa);
(%o1)                           0
(%i2) limit(x+zeroa);
(%o2)                           x

Categories: Constants ·

Constant: zerob ¶

zerob represents an infinitesimal below zero. zerob can be used in expressions. limit simplifies expressions which contain infinitesimals.

See also zeroa and limit.

Categories: Constants ·

5.4.1 Introduction to Lists

Lists are the basic building block for Maxima and Lisp. All data types other than arrays, hashed arrays and 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))

5.4.2 Functions and Variables for Lists

Operator: [ ¶

Operator: ] ¶

[ and ] mark the beginning and end, respectively, of a list.

[ and ] also enclose the subscripts of a list, array, hashed array, or memoizing function. Note that other than for arrays accessing the nth element of a list may need an amount of time that is roughly proportional to n, See Performance considerations for Lists.

Note that if an element of a subscripted variable is written to before a list or an array of this name is declared a hashed array (see Arrays) is created, not a list.

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

Categories: Lists · Operators ·

Function: append (list_1, …, list_n) ¶

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).

See also addrow, addcol and join.

Do example(append); for an example.

Categories: Lists · Expressions ·

Function: assoc
    assoc (key, e, default)
    assoc (key, e) ¶

assoc searches for key as the first part of an argument of e and returns the second part of the first match, if any.

key may be any expression. e must be a nonatomic expression, and every argument of e must have exactly two parts. assoc returns the second part of the first matching argument of e. Matches are determined by is(key = first(a)) where a is an argument of e.

If there are two or more matches, only the first is returned. If there are no matches, default is returned, if specified. Otherwise, false is returned.

See also sublist and member.

Examples:

key may be any expression. e must be a nonatomic expression, and every argument of e must have exactly two parts. assoc returns the second part of the first matching argument of e.

(%i1) assoc (f(x), foo(g(x) = y, f(x) = z + 1, h(x) = 2*u));
(%o1)                         z + 1

If there are two or more matches, only the first is returned.

(%i1) assoc (yy, [xx = 111, yy = 222, yy = 333, yy = 444]);
(%o1)                          222

If there are no matches, default is returned, if specified. Otherwise, false is returned.

(%i1) assoc (abc, [[x, 111], [y, 222], [z, 333]], none); 
(%o1)                         none
(%i2) assoc (abc, [[x, 111], [y, 222], [z, 333]]);
(%o2)                         false

Categories: Lists · Expressions ·

Function: cons
    cons (expr, list)
    cons (expr_1, expr_2) ¶

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 ·

Function: copylist (list) ¶

Returns a copy of the list list.

Categories: Lists ·

Function: create_list (form, x_1, list_1, …, x_n, list_n) ¶

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 ·

Function: delete
    delete (expr_1, expr_2)
    delete (expr_1, expr_2, n) ¶

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))];
(%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 ·

Function: eighth (expr) ¶

Returns the 8th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: endcons
    endcons (expr, list)
    endcons (expr_1, expr_2) ¶

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 ·

Function: fifth (expr) ¶

Returns the 5th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: first (expr) ¶

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.:

(%i1) matrix([1,2],[3,4]);
                                   [ 1  2 ]
(%o1)                              [      ]
                                   [ 3  4 ]
(%i2) first(%);
(%o2)                              [1,2]
(%i3) first(%);
(%o3)                              1
(%i4) first(a*b/c+d+e/x);
                                   a b
(%o4)                              ---
                                    c
(%i5) first(a=b/c+d+e/x);
(%o5)                              a

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. One reason why this may make a difference is that the simplifier re-orders expressions:

(%i1) x+y;
(%o1)                              y+1
(%i2) first(x+y),inflag : true;
(%o2)                              x
(%i3) first(x+y),inflag : false;
(%o3)                              y

The functions second … tenth yield the second through the tenth part of their input argument.

See also firstn and part.

Categories: Lists · Expressions ·

Function: firstn (expr, count) ¶

Returns the first count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

expr may be any nonatomic expression. When expr is something other than a list, firstn returns an expression which has the same operator as expr. count must be a nonnegative integer.

firstn honors the global flag inflag, which governs whether the internal form of an expression is processed (when inflag is true) or the displayed form (when inflag is false).

Note that firstn(expr, 1), which returns a nonatomic expression containing the first argument, is not the same as first(expr), which returns the first argument by itself.

See also lastn and rest.

Examples:

firstn returns the first count elements of expr, if expr has at least count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

(%i2) firstn (mylist, 0);
(%o2)                          []

(%i3) firstn (mylist, 1);
(%o3)                          [1]

(%i4) firstn (mylist, 2);
(%o4)                        [1, a]

(%i5) firstn (mylist, 7);
(%o5)               [1, a, 2, b, 3, x, 4 - y]

firstn returns expr if expr has less than count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

(%i2) firstn (mylist, 100);
(%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

expr may be any nonatomic expression.

(%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))

(%i2) firstn (myfoo, 4);
(%o2)                    foo(1, a, 2, b)

(%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
            m, n

(%i4) firstn (mybar, 4);
(%o4)                  bar    (1, a, 2, b)
                          m, n

(%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $

(%i6) firstn (mymatrix, 3);
                       [ 11  12  13  14 ]
                       [                ]
(%o6)                  [ 21  22  23  24 ]
                       [                ]
                       [ 31  32  33  34 ]

firstn honors the global flag inflag.

(%i1) myexpr : a + b + c + d + e;
(%o1)                   e + d + c + b + a

(%i2) firstn (myexpr, 3), inflag=true;
(%o2)                       c + b + a

(%i3) firstn (myexpr, 3), inflag=false;
(%o3)                       e + d + c

Note that firstn(expr, 1) is not the same as first(expr).

(%i1) firstn ([w, x, y, z], 1);
(%o1)                          [w]

(%i2) first ([w, x, y, z]);
(%o2)                           w

Categories: Lists · Expressions ·

Function: fourth (expr) ¶

Returns the 4th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: join (l, m) ¶

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.

See also append.

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 ·

Function: last (expr) ¶

Returns the last part (term, row, element, etc.) of the expr.

See also lastn.

Categories: Lists · Expressions ·

Function: lastn (expr, count) ¶

Returns the last count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

expr may be any nonatomic expression. When expr is something other than a list, lastn returns an expression which has the same operator as expr. count must be a nonnegative integer.

lastn honors the global flag inflag, which governs whether the internal form of an expression is processed (when inflag is true) or the displayed form (when inflag is false).

Note that lastn(expr, 1), which returns a nonatomic expression containing the last argument, is not the same as last(expr), which returns the last argument by itself.

See also firstn and rest.

Examples:

lastn returns the last count elements of expr, if expr has at least count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

(%i2) lastn (mylist, 0);
(%o2)                          []

(%i3) lastn (mylist, 1);
(%o3)                    [2 z + sin(u)]

(%i4) lastn (mylist, 2);
(%o4)                 [4 - y, 2 z + sin(u)]

(%i5) lastn (mylist, 7);
(%o5)         [a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

lastn returns expr if expr has less than count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

(%i2) lastn (mylist, 100);
(%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

expr may be any nonatomic expression.

(%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))

(%i2) lastn (myfoo, 4);
(%o2)            foo(3, x, 4 - y, 2 z + sin(u))

(%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
            m, n

(%i4) lastn (mybar, 4);
(%o4)          bar    (3, x, 4 - y, 2 z + sin(u))
                  m, n

(%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $

(%i6) lastn (mymatrix, 3);
                     [ 81   82   83   84  ]
                     [                    ]
(%o6)                [ 91   92   93   94  ]
                     [                    ]
                     [ 101  102  103  104 ]

lastn honors the global flag inflag.

(%i1) myexpr : a + b + c + d + e;
(%o1)                   e + d + c + b + a

(%i2) lastn (myexpr, 3), inflag=true;
(%o2)                       e + d + c

(%i3) lastn (myexpr, 3), inflag=false;
(%o3)                       c + b + a

Note that lastn(expr, 1) is not the same as last(expr).

(%i1) lastn ([w, x, y, z], 1);
(%o1)                          [z]

(%i2) last ([w, x, y, z]);
(%o2)                           z

Categories: Lists · Expressions ·

Function: length (expr) ¶

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).

Determining a list’s length typically needs an amount of time proportional to the number of elements in the list. If the length of a list is used inside a loop it therefore might drastically increase the performance if the length is calculated outside the loop instead.

Categories: Lists · Expressions ·

Option variable: listarith ¶

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 ·

Function: listp (expr) ¶

Returns true if expr is a list else false.

Categories: Lists · Predicate functions ·

Function: lreduce
    lreduce (F, s)
    lreduce (F, s, s_0) ¶

Extends the binary function F to an n-ary function by composition, where s is a list.

lreduce(F, s) returns F(... F(F(s_1, s_2), s_3), ... s_n). When the optional argument s_0 is present, the result is equivalent to lreduce(F, cons(s_0, s)).

The function F is first applied to the leftmost list elements, thus the name "lreduce".

See also rreduce, xreduce, and tree_reduce.

Examples:

lreduce without the optional argument.

(%i1) lreduce (f, [1, 2, 3]);
(%o1)                     f(f(1, 2), 3)

(%i2) lreduce (f, [1, 2, 3, 4]);
(%o2)                  f(f(f(1, 2), 3), 4)

lreduce with the optional argument.

(%i1) lreduce (f, [1, 2, 3], 4);
(%o1)                  f(f(f(4, 1), 2), 3)

lreduce applied to built-in binary operators. / is the division operator.

(%i1) lreduce ("^", args ({a, b, c, d}));
                               b c d
(%o1)                       ((a ) )

(%i2) lreduce ("/", args ({a, b, c, d}));
                                a
(%o2)                         -----
                              b c d

Categories: Lists ·

Function: makelist
    makelist ()
    makelist (expr, n)
    makelist (expr, i, i_max)
    makelist (expr, i, i_0, i_max)
    makelist (expr, i, i_0, i_max, step)
    makelist (expr, x, list) ¶

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 jth 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 ·

Function: member (expr_1, expr_2) ¶

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 ·

Function: ninth (expr) ¶

Returns the 9th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: pop (list) ¶

pop removes and returns the first element from the list list. The 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 ·

Function: push (item, list) ¶

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) a: 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) a;
                                     2
(%o9)                  [string, y + x , x]

Categories: Lists · Expressions ·

Function: rest
    rest (expr, n)
    rest (expr) ¶

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 an atom, 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 a 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 ^.

See also firstn and lastn.

(%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 ·

Function: reverse (list) ¶

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.

See also sreverse.

Categories: Lists · Expressions ·

Function: rreduce
    rreduce (F, s)
    rreduce (F, s, s_{n + 1}) ¶

Extends the binary function F to an n-ary function by composition, where s is a list.

rreduce(F, s) returns F(s_1, ... F(s_{n - 2}, F(s_{n - 1}, s_n))). When the optional argument s_{n + 1} is present, the result is equivalent to rreduce(F, endcons(s_{n + 1}, s)).

The function F is first applied to the rightmost list elements, thus the name "rreduce".

See also lreduce, tree_reduce, and xreduce.

Examples:

rreduce without the optional argument.

(%i1) rreduce (f, [1, 2, 3]);
(%o1)                     f(1, f(2, 3))

(%i2) rreduce (f, [1, 2, 3, 4]);
(%o2)                  f(1, f(2, f(3, 4)))

rreduce with the optional argument.

(%i1) rreduce (f, [1, 2, 3], 4);
(%o1)                  f(1, f(2, f(3, 4)))

rreduce applied to built-in binary operators. / is the division operator.

(%i1) rreduce ("^", args ({a, b, c, d}));
                                 d
                                c
                               b
(%o1)                         a

(%i2) rreduce ("/", args ({a, b, c, d}));
                               a c
(%o2)                          ---
                               b d

Categories: Lists ·

Function: second (expr) ¶

Returns the 2nd item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: seventh (expr) ¶

Returns the 7th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: sixth (expr) ¶

Returns the 6th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: sort
    sort (L, P)
    sort (L) ¶

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, sin(1), 1.0b0, %e, %i, %i + 1, inf, 
                                                       minf, 2 x]

Categories: Lists ·

Function: sublist (list, p) ¶

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 ·

Function: sublist_indices (L, P) ¶

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 ·

Function: tenth (expr) ¶

Returns the 10th item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: third (expr) ¶

Returns the 3rd item of expression or list expr. See first for more details.

Categories: Lists · Expressions ·

Function: tree_reduce
    tree_reduce (F, s)
    tree_reduce (F, s, s_0) ¶

Extends the binary function F to an n-ary function by composition, where s is a set or list.

tree_reduce is equivalent to the following: Apply F to successive pairs of elements to form a new list [F(s_1, s_2), F(s_3, s_4), ...], carrying the final element unchanged if there are an odd number of elements. Then repeat until the list is reduced to a single element, which is the return value.

When the optional argument s_0 is present, the result is equivalent tree_reduce(F, cons(s_0, s)).

For addition of floating point numbers, tree_reduce may return a sum that has a smaller rounding error than either rreduce or lreduce.

The elements of s and the partial results may be arranged in a minimum-depth binary tree, thus the name "tree_reduce".

Examples:

tree_reduce applied to a list with an even number of elements.

(%i1) tree_reduce (f, [a, b, c, d]);
(%o1)                  f(f(a, b), f(c, d))

tree_reduce applied to a list with an odd number of elements.

(%i1) tree_reduce (f, [a, b, c, d, e]);
(%o1)               f(f(f(a, b), f(c, d)), e)

Categories: Sets · Lists ·

Function: unique (L) ¶

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]

Function: xreduce
    xreduce (F, s)
    xreduce (F, s, s_0) ¶

Extends the function F to an n-ary function by composition, or, if F is already n-ary, applies F to s. When F is not n-ary, xreduce is the same as lreduce. The argument s is a list.

Functions known to be n-ary include addition +, multiplication *, and, or, max, min, and append. Functions may also be declared n-ary by declare(F, nary). For these functions, xreduce is expected to be faster than either rreduce or lreduce.

When the optional argument s_0 is present, the result is equivalent to xreduce(s, cons(s_0, s)).

Floating point addition is not exactly associative; be that as it may, xreduce applies Maxima’s n-ary addition when s contains floating point numbers.

Examples:

xreduce applied to a function known to be n-ary. F is called once, with all arguments.

(%i1) declare (F, nary);
(%o1)                         done

(%i2) F ([L]) := L;
(%o2)                      F([L]) := L

(%i3) xreduce (F, [a, b, c, d, e]);
(%o3)                    [a, b, c, d, e]

xreduce applied to a function not known to be n-ary. G is called several times, with two arguments each time.

(%i1) G ([L]) := L;
(%o1)                      G([L]) := L

(%i2) xreduce (G, [a, b, c, d, e]);
(%o2)                 [[[[a, b], c], d], e]

(%i3) lreduce (G, [a, b, c, d, e]);
(%o3)                 [[[[a, b], c], d], e]

Categories: Sets · Lists ·

5.4.3 Performance considerations for Lists

Lists provide efficient ways of appending and removing elements. They can be created without knowing their final dimensions. Lisp provides efficient means of copying and handling lists. Also nested lists do not need to be strictly rectangular. These advantages over declared arrays come with the drawback that the amount of time needed for accessing a random element within a list may be roughly proportional to the element’s distance from its beginning. Efficient traversal of lists is still possible, though, by using the list as a stack or a fifo:

(%i1) l:[Test,1,2,3,4];
(%o1)                  [Test, 1, 2, 3, 4]

(%i2) while l # [] do
   disp(pop(l));
                              Test

                                1

                                2

                                3

                                4

(%o2)                         done

Another even faster example would be:

(%i1) l:[Test,1,2,3,4];
(%o1)                  [Test, 1, 2, 3, 4]

(%i2) for i in l do
   disp(pop(l));
                              Test

                                1

                                2

                                3

                                4

(%o2)                         done

Beginning traversal with the last element of a list is possible after reversing the list using reverse (). If the elements of a long list need to be processed in a different order performance might be increased by converting the list into a declared array first.

Note also that the ending condition of for loops is tested for every iteration which means that the result of a length should be cached if it is used in the ending condition:

(%i1) l:makelist(i,i,1,100000)$

(%i2) lngth:length(l);
(%o2)                        100000

(%i3) x:1;
(%o3)                           1

(%i4) for i:1 thru lngth do
    x:x+1$

(%i5) x;
(%o5)                        100001

5.5.1 Functions and Variables for Arrays

Function: array
    array (name, dim_1, …, dim_n)
    array (name, type, dim_1, …, dim_n)
    array ([name_1, …, name_m], dim_1, …, dim_n) ¶

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.

See also arraymake, arrayinfo and make_array.

Categories: Arrays ·

Function: arrayapply (A, [i_1, …, i_n]) ¶

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 ·

Function: arrayinfo (A) ¶

Returns information about the array A. The argument A may be a declared array, a hashed array, a memoizing 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 memoizing 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 array (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 a memoizing 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

Using arrayinfo in order to convert an undeclared array to a declared array:

(%i1) for i:0 thru 10 do a[i]:i^2$

(%i2) indices:map(first,rest(rest(arrayinfo(a))));
(%o2)          [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

(%i3) array(A,fixnum,length(indices)-1)$
(%i4) fillarray(A,map(lambda([x],a[x]),indices))$

(%i5) listarray(A);
(%o5)       [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]

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 ·

Function: arraymake (A, [i_1, …, i_n]) ¶

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)        {Lisp 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)        {Lisp Array: #(1 2 3 4 5 6 7 8 9 10)}

(%i13) arraymake ('A2, [8]);
(%o13)                         A2
                                 8

(%i14) ''%;
(%o14)                          9

Categories: Expressions · Arrays ·

System variable: arrays ¶

Default value: []

arrays is a list of arrays that have been allocated. These comprise arrays declared by array, hashed arrays that can be constructed by implicit definition (assigning something to an element that isn’t yet declared as a list or an array), and memoizing 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)     {Lisp Array: #(NIL NIL NIL NIL NIL NIL NIL)}

(%i5) arrays;
(%o5)                     [aa, bb, cc]

Categories: Arrays · Global variables ·

Function: arraysetapply (A, [i_1, …, i_n], x) ¶

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 ·

Function: fillarray (A, B) ¶

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)           {Lisp Array: #(0 0 0 0 0 0 0 0)}

(%i2) fillarray (a2, [1, 2, 3, 4, 5]);
(%o2)           {Lisp Array: #(1 2 3 4 5 5 5 5)}

(%i3) fillarray (a2, [4]);
(%o3)           {Lisp Array: #(4 4 4 4 4 4 4 4)}

(%i4) fillarray (a2, makelist (i, i, 1, 100));
(%o4)           {Lisp Array: #(1 2 3 4 5 6 7 8)}

Multiple-dimension arrays are filled in row-major order.

(%i1) a3 : make_array (fixnum, 2, 5);
(%o1)      {Lisp 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)      {Lisp Array: #2A((1 2 3 4 5) (6 7 8 9 10))}

(%i3) a4 : make_array (fixnum, 5, 2);
(%o3)   {Lisp Array: #2A((0 0) (0 0) (0 0) (0 0) (0 0))}

(%i4) fillarray (a4, a3);
(%o4)   {Lisp Array: #2A((1 2) (3 4) (5 6) (7 8) (9 10))}

Categories: Arrays ·

Function: listarray (A) ¶

Returns a list of the elements of the array A. The argument A may be an array, an undeclared array (hashed array), a memoizing 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 (hashed arrays), memoizing functions, and subscripted functions, the elements correspond to the index values returned by arrayinfo.

Unbound elements of general arrays (that is, not fixnum and not flonum) are returned as #####. Unbound elements of fixnum or flonum arrays are returned as 0 or 0.0, respectively. Unbound elements of hashed arrays, memoizing 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 array (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 a memoizing 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 ·

Function: make_array (type, dim_1, …, dim_n) ¶

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)         {Lisp Array: #(0 0 0 0 0 0 0 0 0 0)}

(%i2) A1 [8] : 1729;
(%o2)                         1729

(%i3) A1;
(%o3)        {Lisp Array: #(0 0 0 0 0 0 0 0 1729 0)}

(%i4) A2 : make_array (flonum, 10);
(%o4) {Lisp 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) 
 {Lisp 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) {Lisp Array: #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)}

(%i8) A3 [4] : x - y - z;
(%o8)                     (- z) - y + x

(%i9) A3;
(%o9) {Lisp Array: #(NIL NIL NIL NIL
               ((MPLUS SIMP) $X ((MTIMES SIMP) -1 $Y) ((MTIMES S\
IMP) -1 $Z))
               NIL NIL NIL NIL NIL)}

(%i10) A4 : make_array (fixnum, 2, 3, 5);
(%o10) {Lisp 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) {Lisp Array: #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 1\
5))
                 ((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29\
 30)))}

(%i12) A4 [0, 2, 1];
(%o12)                         12

Categories: Arrays ·

Function: rearray (A, dim_1, …, dim_n) ¶

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 ·

Function: remarray
    remarray (A_1, …, A_n)
    remarray (all) ¶

Removes arrays and array associated functions and frees the storage occupied. The arguments may be declared arrays, 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 clear the cache of a memoizing function.

remarray returns the list of arrays removed.

remarray quotes its arguments.

Categories: Arrays ·

Function: subvar (x, i) ¶

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 ·

Function: subvarp (expr) ¶

Returns true if expr is a subscripted variable, for example a[i].

Categories: Predicate functions ·

Option variable: use_fast_arrays ¶

Default value: false

When use_fast_arrays is true, arrays declared by array are values instead of properties, and undeclared arrays (hashed arrays) are implemented as Lisp hashed arrays.

When use_fast_arrays is false, arrays declared by array are properties, and undeclared arrays are implemented with Maxima’s own hashed array implementation.

Note that the code use_fast_arrays switches to is not necessarily faster than the default one; Arrays created by make_array are not affected by use_fast_arrays.

See also translate_fast_arrays.

Categories: Arrays · Global flags ·

Option variable: translate_fast_arrays ¶

Default value: false

When translate_fast_arrays is true, the Maxima-to-Lisp translator generates code that assumes arrays are values instead of properties, as if use_fast_arrays were true.

When translate_fast_arrays is false, the Maxima-to-Lisp translator generates code that assumes arrays are properties, as if use_fast_arrays were false.

Categories: Arrays · Translation flags and variables ·

5.6.1 Introduction to Structures

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.

5.6.2 Functions and Variables for Structures

Global variable: structures ¶

structures is the list of user-defined structures defined by defstruct.

Categories: Structures · Global variables ·

Function: defstruct
    defstruct (S(a_1, …, a_n))
    defstruct (S(a_1 = v_1, …, a_n = v_n)) ¶

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 ·