math

package math

The math package contains classes, traits and objects for common mathematical operations. Its package object defines exponentiation, logarithmic, trigonometric, etc. operators and functions.

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case class Complex(re: Double, im: Double = 0.0) extends Fractional[Complex] with Ordered[Complex] with Product with Serializable
The Complex class is used to represent and operate on complex numbers.
The Complex class is used to represent and operate on complex numbers. Internally, a complex number is represented as two double precision floating point numbers (Double). Externally, two forms are supported:
a+bi = 2.1+3.2i via: Complex ("2.1+3.2i"), 'toString' (a, b) = (2.1, 3.2) via: create ("(2.1, 3.2)"), 'toString2'
Note: 'i * i = -1'. -------------------------------------------------------------------------
re
the real part (e.g., 2.1)
im
the imaginary part (e.g., 3.2)
final class Double_Exp extends AnyVal
The Double_Exp value class adds an exponentiation operator 'x ~^{y' and
a 'near_eq' operator 'x =~ y' to Double.
The '~}' has higher precedence than '*' or '/'.
type FunctionS2S = (Double) ⇒ Double
The type definition for a function of a scalar (f: Double => Double)
The type definition for a function of a scalar (f: Double => Double)

See also
also scalation.linalgebra
type Functions = Array[FunctionS2S]
The type definition for an array of scalar functions
final class Int_Exp extends AnyVal
The Int_Exp value class adds an exponentiation operator 'x ~^{y' and a
'near_eq' operator 'x =~ y' to Int.
The '~}' has higher precedence than '*' or '/'.
final class Long_Exp extends AnyVal
The Long_Exp value class adds an exponentiation operator 'x ~^{y' and a
'near_eq' operator 'x =~ y' to Long.
The '~}' has higher precedence than '*' or '/'.
case class ProbNum(x: Double, p: Double = 1.0) extends Numeric[ProbNum] with Ordered[ProbNum] with Error with Product with Serializable
The ProbNum class is used to represent probabilistic numbers '(x, p)' where 'x' is a real number and 'p' is its probability of occurrence.
The ProbNum class is used to represent probabilistic numbers '(x, p)' where 'x' is a real number and 'p' is its probability of occurrence. FIX: Currently this class is half-baked!!!
x
the real number (double precision)
p
the probability of its occurrence [0, 1]

See also
http://okmij.org/ftp/Computation/monads.html#random-var-monad
case class Rational(num: Long, den: Long = 1l) extends Fractional[Rational] with Ordered[Rational] with Product with Serializable
The Rational class is used to represent and operate on rational numbers.
The Rational class is used to represent and operate on rational numbers. Internally, a rational number is represented as two long integers. Externally, two forms are supported:
a/b = 2/3 via: Rational ("2/3"), 'toString' (a, b) = (2, 3) via: create ("(2, 3)") 'toString2'
Rational number can be created without loss of precision using the constructor, 'apply', 'create' or 'fromBigDecimal' methods. Other methods may lose precision.
num
the numerator (e.g., 2)
den
the denominator (e.g., 3)
case class Real(hi: Double, lo: Double = 0.0) extends Fractional[Real] with Ordered[Real] with Product with Serializable
The Real class provides higher precision floating point numbers using the Double Double technique, which supports 106 bits of precision.
The Real class provides higher precision floating point numbers using the Double Double technique, which supports 106 bits of precision. ----------------------------------------------------------------------------- Code adapted from DoubleDouble.java:
hi
the high portion of the real number
lo
the low portion of the real number

See also
oai.cwi.nl/oai/asset/9159/9159A.pdf -----------------------------------------------------------------------------
http://tsusiatsoftware.net/dd/main.html

Value Members

val THRES: Double
The threshold used for near equality
def cot(x: Double): Double
Return the cotangent of 'x'
def csc(x: Double): Double
Return the cosecant of 'x'
implicit def double_exp(x: Double): Double_Exp
Implicit conversion from 'Double' to 'Double_Exp', which supports exponentiation and nearly equals.
Implicit conversion from 'Double' to 'Double_Exp', which supports exponentiation and nearly equals.
x
the base parameter
implicit def int_exp(x: Int): Int_Exp
Implicit conversion from 'Int' to 'Int_Exp', which supports exponentiation and nearly equals.
Implicit conversion from 'Int' to 'Int_Exp', which supports exponentiation and nearly equals.
x
the base parameter
def log2(x: Double): Double
Compute the log of 'x' base 2.
Compute the log of 'x' base 2.
x
the value whose log is sought
val log_10: Double
The natural log of 10
val log_2: Double
The natural log of 2
def logb(b: Double, x: Double): Double
Compute the log of 'x' base 'b'.
Compute the log of 'x' base 'b'.
b
the base of the logarithm
x
the value whose log is sought
implicit def long_exp(x: Long): Long_Exp
Implicit conversion from 'Long' to 'Long_Exp', which supports exponentiation and nearly equals.
Implicit conversion from 'Long' to 'Long_Exp', which supports exponentiation and nearly equals.
x
the base parameter
def near_eq(x: Double, y: Double): Boolean
Determine whether two double precision floating point numbers 'x' and 'y' are nearly equal.
Determine whether two double precision floating point numbers 'x' and 'y' are nearly equal. Two numbers are considered to be nearly equal, if within '2 EPSILON'. A number is considered to be nearly zero, if within '2 MIN_NORMAL'. To accommodate round-off errors, may use 'TOL' instead of 'EPSILON'. --------------------------------------------------------------------------
x
the first double precision floating point number
y
the second double precision floating point number

See also
http://stackoverflow.com/questions/10034149/why-is-nan-not-equal-to-nan --------------------------------------------------------------------------
stackoverflow.com/questions/4915462/how-should-i-do-floating-point-comparison -------------------------------------------------------------------------- If both 'x' and 'y' are NaN (Not-a-Number), the IEEE standard indicates that should be considered always not equal. For 'near_eq', they are considered nearly equal. Comment out the first line below to conform more closely to the IEEE standard.
BasicTest
def nexp(x: Double): Double
Negative exponential function (e to the minus 'x').
Negative exponential function (e to the minus 'x').
x
the argument of the function
val noComplex: Complex
Missing value representation (use null if available, otherwise most negative value)
val noDouble: Double
val noInt: Int
val noLong: Long
val noRational: Rational
val noReal: Real
val noStrNum: StrNum
val noTimeNum: TimeNum
def oneIf(cond: Boolean): Int
Return 1 if condition 'cond' is true, else 0.
Return 1 if condition 'cond' is true, else 0.
cond
the condition to evaluate
def pow(x: Long, y: Long): Long
Power function for scala Longs 'x ~^ y'. Compute: 'math.pow (x, y).toLong' without using floating point, so as to not lose precision.
Power function for scala Longs 'x ~^ y'. Compute: 'math.pow (x, y).toLong' without using floating point, so as to not lose precision.
x
the Long base parameter
y
the Long exponent parameter
def root(x: Long, y: Long): Long
Find the 'y'-th root of 'x', i.e., 'x ~^{1/y' for scala Longs.
'r = x ~} 1/y' is largest long integer 'r' such that 'r ~^ y <= x'.
Find the 'y'-th root of 'x', i.e., 'x ~^{1/y' for scala Longs.
'r = x ~} 1/y' is largest long integer 'r' such that 'r ~^ y <= x'.
x
the Long base parameter
y
the Long root level (reciprocal exponent) parameter

See also
http://stackoverflow.com/questions/8826822/calculate-nth-root-with-integer-arithmetic
http://en.wikipedia.org/wiki/Shifting_nth_root_algorithm
def roundTo(x: Double, places: Int = 4): Double
Round the given double to the desired number of decimal places.
Round the given double to the desired number of decimal places.
x
the double number which needs to be rounded off
places
the desired number of decimal places
def sec(x: Double): Double
Return the secant of 'x'
def sign(x: Double, y: Double): Double
Return the absolute value of 'x' with the sign of 'y'.
Return the absolute value of 'x' with the sign of 'y'.
x
the value contributor
y
the sign contributor
def sq(x: Double): Double
Compute the square of 'x'.
Compute the square of 'x'.
x
the number to sqaure
object Combinatorics extends Error
The Combinatorics object provides several common combinatorics functions, such as factorial permutations, combinations, gamma and beta functions.
object CombinatoricsTest extends App
The CombinatoricsTest object tests the methods in the Combinatorics object.
object CombinatoricsTest2 extends App
The CombinatoricsTest2 object tests the Gamma and factorial methods in the Combinatorics object.
object Complex extends Serializable
The Complex companion object defines the origin (zero) and the fourth roots of unity as well as some utility functions.
object ComplexTest extends App
The ComplexTest object is used to test the Complex class.
The ComplexTest object is used to test the Complex class. > runMain scalation.math.ComplexTest
object ExtensionTest extends App
The ExtensionTest object is used to test the Int_Exp, Long_Exp and Double_Exp classes.
The ExtensionTest object is used to test the Int_Exp, Long_Exp and Double_Exp classes. > runMain scalation.math.ExtensionTest
object ExtremeD
The ExtremeD object contains constants representing extreme values for Double (IEEE 754 double precision floating point numbers).
The ExtremeD object contains constants representing extreme values for Double (IEEE 754 double precision floating point numbers).

See also
en.wikipedia.org/wiki/Double-precision_floating-point_format ------------------------------------------------------------------------------ 64 bits: 1 sign-bit, 11 exponent-bits, 52 mantissa-bits + 1 implicit
http://docs.oracle.com/javase/8/docs/api/java/lang/Double.html
object ExtremeDTest extends App
The ExtremeDTest object is used to test the ExtremeD class.
The ExtremeDTest object is used to test the ExtremeD class. > runMain scalation.math.ExtremeDTest
object MathTest extends App
The MathTest object is used to test the supplemental power, root, exponential, sign, indicator, logarithmic and trigonmetric functions defined in the 'scalation.math' package object.
The MathTest object is used to test the supplemental power, root, exponential, sign, indicator, logarithmic and trigonmetric functions defined in the 'scalation.math' package object. > runMain scalation.math.MathTest
object Primes
The Primes object provides an array of 1000 prime numbers as well as methods to generate prime numbers within a given range.
object PrimesTest extends App
The PrimesTest object is use to perform timing test on the Primes object.
object ProbNumTest extends App
The ProbNumTest object is used to test the ProbNum class.
The ProbNumTest object is used to test the ProbNum class. > runMain scalation.math.ProbNumTest
object Rational extends Serializable
The Rational companion object defines the origin (zero), one and minus one as well as some utility functions.
object RationalTest extends App
The RationalTest object is used to test the Rational class.
object Real extends Serializable
The Real companion object defines the origin (zero), one half and one as well as some utility functions.
object RealTest extends App
The RealTest object is used to test the Real class.
The RealTest object is used to test the Real class. > runMain scalation.math.RealTest
object StrNumTest extends App
The StrNumTest object is used to test the StrNum class.
The StrNumTest object is used to test the StrNum class. > runMain scalation.math.StrNumTest
object StrO
The StrO object is used to represent and operate on string numbers.
The StrO object is used to represent and operate on string numbers. It contains an implicit class definition for StrNum, which extends strings with Numeric operations, it it can be used in VectorS. The semantics of StrNum operators are similar those in Pike.

See also
http://docs.roxen.com/pike/7.0/tutorial/strings/operators.xml
object TimeNumTest extends App
The TimeNumTest object is used to test the TimeNum class.
The TimeNumTest object is used to test the TimeNum class. > runMain scalation.math.TimeNumTest
object TimeNumTest2 extends App
The TimeNumTest2 object is used to test the TimeNum class regarding approximately equal date-times.
The TimeNumTest2 object is used to test the TimeNum class regarding approximately equal date-times. > runMain scalation.math.TimeNumTest2
object TimeNumTest3 extends App
The TimeNumTest3 object is used to test the TimeNum class regarding its getChrono method.
The TimeNumTest3 object is used to test the TimeNum class regarding its getChrono method. FIX - crashes - format exception > runMain scalation.math.TimeNumTest3
object TimeO
The TimeO object is used to represent and operate on date-time values.
The TimeO object is used to represent and operate on date-time values. It contains an implicit class definition for TimeNum, which extends Instant with Numeric operations that can be used in VectorT. The java.time.Instant class stores nano-seconds since the UNIX Epoch using a long for seconds and int for nano-seconds (12 bytes or 96 bits). The java.time.ZonedDateTime class is used with it to handle various date-time formats.

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