+, -, *, /, ^
,
Div, Mod
,
Gcd
,
Lcm
,
<<, >>
,
FromBase, ToBase
,
Precision
,
GetPrecision
,
N
,
Rationalize
,
IntLog
,
IntNthRoot
,
NthRoot
,
ContFrac
,
ContFracList, ContFracEval
,
GuessRational, NearRational, BracketRational
,
Decimal
,
TruncRadian
,
Floor
,
Ceil
,
Round
,
Pslq
.
Arithmetic and other operations on numbers
Besides the usual arithmetical operations,
Yacas defines some more advanced operations on
numbers. Many of them also work on polynomials.
+, -, *, /, ^
|
arithmetic operations |
Div, Mod
|
division with remainder |
Gcd
|
greatest common divisor |
Lcm
|
least common multiple |
<<, >>
|
shift operators |
FromBase, ToBase
|
conversion from/to non-decimal base |
Precision
|
set the precision |
GetPrecision
|
get the current precision |
N
|
compute numerical approximation |
Rationalize
|
convert floating point numbers to fractions |
IntLog
|
integer part of logarithm |
IntNthRoot
|
integer part of n-th root |
NthRoot
|
calculate/simplify nth root of an integer |
ContFrac
|
continued fraction expansion |
ContFracList, ContFracEval
|
manipulate continued fractions |
GuessRational, NearRational, BracketRational
|
find optimal rational approximations |
Decimal
|
decimal representation of a rational |
TruncRadian
|
remainder modulo 2*Pi |
Floor
|
round a number downwards |
Ceil
|
round a number upwards |
Round
|
round a number to the nearest integer |
Pslq
|
search for integer relations between reals |
+, -, *, /, ^ -- arithmetic operations
Standard library
Calling format:
x+y (precedence 6)
+x
x-y (precedence 5)
-x
x*y (precedence 3)
x/y (precedence 3)
x^y (precedence 2)
|
Parameters:
x and y -- objects for which arithmetic operations are defined
Description:
These are the basic arithmetic operations. They can work on integers,
rational numbers, complex numbers, vectors, matrices and lists.
These operators are implemented in the standard math library (as opposed
to being built-in). This means that they can be extended by the user.
Examples:
In> 2+3
Out> 5;
In> 2*3
Out> 6;
|
Div, Mod -- division with remainder
Standard library
Calling format:
Parameters:
x, y -- integers or univariate polynomials
Description:
Div performs integer division and Mod returns the remainder after division. Div and
Mod are also defined for polynomials.
If Div(x,y) returns "a" and Mod(x,y) equals "b", then these numbers satisfy x=a*y+b and 0<=b<y.
Examples:
In> Div(5,3)
Out> 1;
In> Mod(5,3)
Out> 2;
|
See also:
Gcd
,
Lcm
.
Gcd -- greatest common divisor
Standard library
Calling format:
Parameters:
n, m -- integers or Gaussian integers or univariate polynomials
list -- a list of all integers or all univariate polynomials
Description:
This function returns the greatest common divisor of "n" and "m".
The gcd is the largest number that divides "n" and "m". It is
also known as the highest common factor (hcf). The library code calls
MathGcd, which is an internal function. This
function implements the "binary Euclidean algorithm" for determining the
greatest common divisor:
Routine for calculating Gcd(n,m)
- if n=m then return n
- if both n and m are even then return 2*Gcd(n/2,m/2)
- if exactly one of n or m (say n) is even then return Gcd(n/2,m)
- if both n and m are odd and, say, n>m then return Gcd((n-m)/2,m)
This is a rather fast algorithm on computers that can efficiently shift
integers. When factoring Gaussian integers, a slower recursive algorithm is used.
If the second calling form is used, Gcd will
return the greatest common divisor of all the integers or polynomials
in "list". It uses the identity
Gcd(a,b,c)=Gcd(Gcd(a,b),c).
Examples:
In> Gcd(55,10)
Out> 5;
In> Gcd({60,24,120})
Out> 12;
In> Gcd( 7300 + 12*I, 2700 + 100*I)
Out> Complex(-4,4);
|
See also:
Lcm
.
Lcm -- least common multiple
Standard library
Calling format:
Parameters:
n, m -- integers or univariate polynomials
list -- list of integers
Description:
This command returns the least common multiple of "n" and "m" or all of
the integers in the list list.
The least common multiple of two numbers "n" and "m" is the lowest
number which is an integer multiple of both "n" and "m".
It is calculated with the formula
Lcm(n,m)=Div(n*m,Gcd(n,m)).
This means it also works on polynomials, since Div, Gcd and multiplication are also defined for
them.
Examples:
In> Lcm(60,24)
Out> 120;
In> Lcm({3,5,7,9})
Out> 315;
|
See also:
Gcd
.
<<, >> -- shift operators
Standard library
Calling format:
Parameters:
n, m -- integers
Description:
These operators shift integers to the left or to the right.
They are similar to the C shift operators. These are sign-extended
shifts, so they act as multiplication or division by powers of 2.
Examples:
In> 1 << 10
Out> 1024;
In> -1024 >> 10
Out> -1;
|
FromBase, ToBase -- conversion from/to non-decimal base
Internal function
Calling format:
FromBase(base,number)
ToBase(base,number)
|
Parameters:
base -- integer, base to write the numbers in
number -- integer, number to write out in the base representation
Description:
FromBase converts "number", written in base
"base", to base 10. ToBase converts "number",
written in base 10, to base "base".
These functions use the p-adic expansion capabilities of the built-in
arbitrary precision math libraries.
Examples:
In> FromBase(2,111111)
Out> 63;
In> ToBase(16,255)
Out> ff;
|
The first command writes the binary number 111111
in decimal base. The second command converts 255
(in decimal base) to hexadecimal base.
See also:
PAdicExpand
.
Precision -- set the precision
Internal function
Calling format:
Parameters:
n -- integer, new precision
Description:
This command sets the number of binary digits to be used in
calculations. All subsequent floating point operations will allow for
at least n digits after the decimal point.
When the precision is changed, all variables containing previously calculated values
remain unchanged.
The Precision function only makes all further calculations proceed with a different precision.
Examples:
In> Precision(10)
Out> True;
In> N(Sin(1))
Out> 0.8414709848;
In> Precision(20)
Out> True;
In> x:=N(Sin(1))
Out> 0.84147098480789650665;
In> GetPrecision()
Out> 20;
In> [ Precision(10); x; ]
Out> 0.84147098480789650665;
In> x+0
Out> 0.8414709848;
|
See also:
GetPrecision
,
N
.
GetPrecision -- get the current precision
Internal function
Calling format:
Description:
This command returns the current precision, as set by Precision.
Examples:
In> GetPrecision();
Out> 10;
In> Precision(20);
Out> True;
In> GetPrecision();
Out> 20;
|
See also:
Precision
,
N
.
N -- compute numerical approximation
Standard library
Calling format:
Parameters:
expr -- expression to evaluate
prec -- integer, precision to use
Description:
This function forces Yacas to give a numerical approximation to the
expression "expr", using "prec" digits if the second calling
sequence is used, and the precision as set by SetPrecision otherwise. This overrides the normal
behaviour, in which expressions are kept in symbolic form (eg. Sqrt(2) instead of 1.41421).
Application of the N operator will make Yacas
calculate floating point representations of functions whenever
possible. In addition, the variable Pi is bound to
the value of Pi calculated at the current precision.
(This value is a "cached constant", so it is not recalculated each time N is used, unless the precision is increased.)
Examples:
In> 1/2
Out> 1/2;
In> N(1/2)
Out> 0.5;
In> Sin(1)
Out> Sin(1);
In> N(Sin(1),10)
Out> 0.8414709848;
In> Pi
Out> Pi;
In> N(Pi,20)
Out> 3.14159265358979323846;
|
See also:
Precision
,
GetPrecision
,
Pi
,
CachedConstant
.
Rationalize -- convert floating point numbers to fractions
Standard library
Calling format:
Parameters:
expr -- an expression containing real numbers
Description:
This command converts every real number in the expression "expr"
into a rational number. This is useful when a calculation needs to be
done on floating point numbers and the algorithm is unstable.
Converting the floating point numbers to rational numbers will force
calculations to be done with infinite precision (by using rational
numbers as representations).
It does this by finding the smallest integer n such that multiplying
the number with 10^n is an integer. Then it divides by 10^n again,
depending on the internal gcd calculation to reduce the resulting
division of integers.
Examples:
In> {1.2,3.123,4.5}
Out> {1.2,3.123,4.5};
In> Rationalize(%)
Out> {6/5,3123/1000,9/2};
|
See also:
IsRational
.
IntLog -- integer part of logarithm
Standard library
Calling format:
Parameters:
n, base -- positive integers
Description:
IntLog calculates the integer part of the logarithm of n in base base. The algorithm uses only integer math and may be faster than computing
Ln(n)/Ln(base)
with multiple precision floating-point math and rounding off to get the integer part.
This function can also be used to quickly count the digits in a given number.
Examples:
Count the number of bits:
In> IntLog(257^8, 2)
Out> 64;
|
Count the number of decimal digits:
In> IntLog(321^321, 10)
Out> 804;
|
See also:
IntNthRoot
,
Div
,
Mod
,
Ln
.
IntNthRoot -- integer part of n-th root
Standard library
Calling format:
Parameters:
x, n -- positive integers
Description:
IntNthRoot calculates the integer part of the n-th root of x. The algorithm uses only integer math and may be faster than computing x^(1/n) with floating-point and rounding.
This function is used to test numbers for prime powers.
Example:
In> IntNthRoot(65537^111, 37)
Out> 281487861809153;
|
See also:
IntLog
,
MathPower
,
IsPrimePower
.
NthRoot -- calculate/simplify nth root of an integer
Standard library
Calling format:
NthRoot(m,n)
Parameters:
m -- a non-negative integer (m>0)
n -- a positive integer greater than 1 ( n>1)
Description:
NthRoot(m,n) calculates the integer part of the n-th root m^(1/n) and
returns a list {f,r}. f and r are both positive integers
that satisfy f^n*r= m.
In other words, f is the largest integer such that m divides f^n and r is the remaining factor.
For large m and small n
NthRoot may work quite slowly. Every result {f,r} for given
m, n is saved in a lookup table, thus subsequent calls to
NthRoot with the same values m, n will be executed quite
fast.
Example:
In> NthRoot(12,2)
Out> {2,3};
In> NthRoot(81,3)
Out> {3,3};
In> NthRoot(3255552,2)
Out> {144,157};
In> NthRoot(3255552,3)
Out> {12,1884};
|
See also:
IntNthRoot
,
Factors
,
MathPower
.
ContFrac -- continued fraction expansion
Standard library
Calling format:
ContFrac(x)
ContFrac(x, depth)
|
Parameters:
x -- number or polynomial to expand in continued fractions
depth -- integer, maximum required depth of result
Description:
This command returns the continued fraction expansion of x, which
should be either a floating point number or a polynomial. If
depth is not specified, it defaults to 6. The remainder is
denoted by rest.
This is especially useful for polynomials, since series expansions
that converge slowly will typically converge a lot faster if
calculated using a continued fraction expansion.
Examples:
In> PrettyForm(ContFrac(N(Pi)))
1
--------------------------- + 3
1
----------------------- + 7
1
------------------ + 15
1
-------------- + 1
1
-------- + 292
rest + 1
|
Out> True;
In> PrettyForm(ContFrac(x^2+x+1, 3))
x
---------------- + 1
x
1 - ------------
x
-------- + 1
rest + 1
Out> True;
|
See also:
ContFracList
,
NearRational
,
GuessRational
,
PAdicExpand
,
N
.
ContFracList, ContFracEval -- manipulate continued fractions
Standard library
Calling format:
ContFracList(frac)
ContFracList(frac, depth)
ContFracEval(list)
ContFracEval(list, rest)
|
Parameters:
frac -- a number to be expanded
depth -- desired number of terms
list -- a list of coefficients
rest -- expression to put at the end of the continued fraction
Description:
The function ContFracList computes terms of the continued fraction
representation of a rational number frac. It returns a list of terms of length depth. If depth is not specified, it returns all terms.
The function ContFracEval converts a list of coefficients into a continued fraction expression. The optional parameter rest specifies the symbol to put at the end of the expansion. If it is not given, the result is the same as if rest=0.
Examples:
In> A:=ContFracList(33/7 + 0.000001)
Out> {4,1,2,1,1,20409,2,1,13,2,1,4,1,1,3,3,2};
In> ContFracEval(Take(A, 5))
Out> 33/7;
In> ContFracEval(Take(A,3), remainder)
Out> 1/(1/(remainder+2)+1)+4;
|
See also:
ContFrac
,
GuessRational
.
GuessRational, NearRational, BracketRational -- find optimal rational approximations
Standard library
Calling format:
GuessRational(x)
GuessRational(x, digits)
NearRational(x)
NearRational(x, digits)
BracketRational(x, eps)
|
Parameters:
x -- a number to be approximated (must be already evaluated to floating-point)
digits -- desired number of decimal digits (integer)
eps -- desired precision
Description:
The functions GuessRational(x) and NearRational(x) attempt to find "optimal"
rational approximations to a given value x. The approximations are "optimal"
in the sense of having smallest numerators and denominators among all rational
numbers close to x. This is done by computing a continued fraction
representation of x and truncating it at a suitably chosen term. Both
functions return a rational number which is an approximation of x.
Unlike the function Rationalize() which converts floating-point numbers to
rationals without loss of precision, the functions GuessRational() and
NearRational() are intended to find the best rational that is approximately
equal to a given value.
The function GuessRational() is useful if you have obtained a
floating-point representation of a rational number and you know
approximately how many digits its exact representation should contain.
This function takes an optional second parameter digits which limits
the number of decimal digits in the denominator of the resulting
rational number. If this parameter is not given, it defaults to half
the current precision. This function truncates the continuous fraction
expansion when it encounters an unusually large value (see example).
This procedure does not always give the "correct" rational number; a
rule of thumb is that the floating-point number should have at least as
many digits as the combined number of digits in the numerator and the
denominator of the correct rational number.
The function NearRational(x) is useful if one needs to
approximate a given value, i.e. to find an "optimal" rational number
that lies in a certain small interval around a certain value x. This
function takes an optional second parameter digits which has slightly
different meaning: it specifies the number of digits of precision of
the approximation; in other words, the difference between x and the
resulting rational number should be at most one digit of that
precision. The parameter digits also defaults to half of the current
precision.
The function BracketRational(x,eps) can be used to find approximations with a given relative precision from above and from below.
This function returns a list of two rational numbers {r1,r2} such that r1<x<r2 and Abs(r2-r1)<Abs(x*eps).
The argument x must be already evaluated to enough precision so that this approximation can be meaningfully found.
If the approximation with the desired precision cannot be found, the function returns an empty list.
Examples:
Start with a rational number and obtain a floating-point approximation:
In> x:=N(956/1013)
Out> 0.9437314906
In> Rationalize(x)
Out> 4718657453/5000000000;
In> V(GuessRational(x))
GuessRational: using 10 terms of the
continued fraction
Out> 956/1013;
In> ContFracList(x)
Out> {0,1,16,1,3,2,1,1,1,1,508848,3,1,2,1,2,2};
|
The first 10 terms of this continued fraction correspond to the correct continued fraction for the original rational number.
In> NearRational(x)
Out> 218/231;
|
This function found a different rational number closeby because the precision was not high enough.
In> NearRational(x, 10)
Out> 956/1013;
|
Find an approximation to Ln(10) good to 8 digits:
In> BracketRational(N(Ln(10)), 10^(-8))
Out> {12381/5377,41062/17833};
|
See also:
ContFrac
,
ContFracList
,
Rationalize
.
Decimal -- decimal representation of a rational
Standard library
Calling format:
Parameters:
frac -- a rational number
Description:
This function returns the infinite decimal representation of a
rational number frac. It returns a list, with the first element
being the number before the decimal point and the last element the
sequence of digits that will repeat forever. All the intermediate list
elements are the initial digits before the period sets in.
Examples:
In> Decimal(1/22)
Out> {0,0,{4,5}};
In> N(1/22,30)
Out> 0.045454545454545454545454545454;
|
See also:
N
.
TruncRadian -- remainder modulo 2*Pi
Standard library
Calling format:
Parameters:
r -- a number
Description:
TruncRadian calculates Mod(r,2*Pi), returning a value between 0
and 2*Pi. This function is used in the trigonometry functions, just
before doing a numerical calculation using a Taylor series. It greatly
speeds up the calculation if the value passed is a large number.
The library uses the formula
TruncRadian(r)=r-Floor(r/(2*Pi))*2*Pi,
where r and 2*Pi are calculated with twice the precision used in the
environment to make sure there is no rounding error in the significant
digits.
Examples:
In> 2*Pi()
Out> 6.283185307;
In> TruncRadian(6.28)
Out> 6.28;
In> TruncRadian(6.29)
Out> 0.0068146929;
|
See also:
Sin
,
Cos
,
Tan
.
Floor -- round a number downwards
Standard library
Calling format:
Parameters:
x -- a number
Description:
This function returns Floor(x), the largest integer smaller than or equal to x.
Examples:
In> Floor(1.1)
Out> 1;
In> Floor(-1.1)
Out> -2;
|
See also:
Ceil
,
Round
.
Ceil -- round a number upwards
Standard library
Calling format:
Parameters:
x -- a number
Description:
This function returns Ceil(x), the smallest integer larger than or equal to x.
Examples:
In> Ceil(1.1)
Out> 2;
In> Ceil(-1.1)
Out> -1;
|
See also:
Floor
,
Round
.
Round -- round a number to the nearest integer
Standard library
Calling format:
Parameters:
x -- a number
Description:
This function returns the integer closest to x. Half-integers
(i.e. numbers of the form n+0.5, with n an integer) are
rounded upwards.
Examples:
In> Round(1.49)
Out> 1;
In> Round(1.51)
Out> 2;
In> Round(-1.49)
Out> -1;
In> Round(-1.51)
Out> -2;
|
See also:
Floor
,
Ceil
.
Pslq -- search for integer relations between reals
Standard library
Calling format:
Parameters:
xlist -- list of numbers
precision -- required number of digits precision of calculation
Description:
This function is an integer relation detection algorithm. This means
that, given the numbers x[i] in the list "xlist", it tries
to find integer coefficients a[i] such that
a[1]*x[1] + ... + a[n]*x[n]=0.
The list of integer coefficients is returned.
The numbers in "xlist" must evaluate to floating point numbers if
the N operator is applied on them.
Example:
In> Pslq({ 2*Pi+3*Exp(1), Pi, Exp(1) },20)
Out> {1,-2,-3};
|
Note: in this example the system detects correctly that
1*(2*Pi+3*e)+(-2)*Pi+(-3)*e=0.
See also:
N
.