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  1. case class Above [A] extends Interval[A] with Product with Serializable
  2. final class Algebraic extends ScalaNumber with ScalaNumericConversions with Serializable

    Algebraic provides an exact number type for algebraic numbers.

    Algebraic provides an exact number type for algebraic numbers. Algebraic numbers are roots of polynomials with rational coefficients. With it, we can represent expressions involving addition, multiplication, division, n-roots (eg. sqrt or cbrt), and roots of rational polynomials. So, it is similar Rational, but adds roots as a valid, exact operation. The cost is that this will not be as fast as Rational for many operations.

    In general, you can assume all operations on this number type are exact, except for those that explicitly construct approximations to an Algebraic number, such as toBigDecimal.

    For an overview of the ideas, algorithms, and proofs of this number type, you can read the following papers:

    • "On Guaranteed Accuracy Computation." C. K. Yap.
    • "Recent Progress in Exact Geometric Computation." C. Li, S. Pion, and C. K. Yap.
    • "A New Constructive Root Bound for Algebraic Expressions" by C. Li & C. K. Yap.
    • "A Separation Bound for Real Algebraic Expressions." C. Burnikel, et al.
    Annotations
    @SerialVersionUID()
  3. class AlgebraicAlgebra extends AlgebraicIsFieldWithNRoot with AlgebraicIsReal with Serializable
    Annotations
    @SerialVersionUID()
  4. trait AlgebraicInstances extends AnyRef
  5. case class All [A] extends Interval[A] with Product with Serializable
  6. case class Below [A] extends Interval[A] with Product with Serializable
  7. abstract class BinaryMerge extends AnyRef

    Abstract class that can be used to implement custom binary merges with e.g.

    Abstract class that can be used to implement custom binary merges with e.g. special collision behavior or an ordering that is not defined via an Order[T] typeclass

  8. trait BitString [A] extends Bool[A]
  9. case class Bounded [A] extends Interval[A] with Product with Serializable
  10. final case class Complex [T](real: T, imag: T) extends ScalaNumber with ScalaNumericConversions with Serializable with Product
    Annotations
    @SerialVersionUID()
  11. trait ComplexInstances extends ComplexInstances1
  12. trait ComplexInstances0 extends AnyRef
  13. trait ComplexInstances1 extends ComplexInstances0
  14. class ComplexIsNumeric [A] extends ComplexEq[A] with ComplexIsField[A] with Numeric[Complex[A]] with ComplexIsTrig[A] with ComplexIsNRoot[A] with ConvertableFromComplex[A] with ConvertableToComplex[A] with Order[Complex[A]] with ComplexIsSigned[A] with Serializable
    Annotations
    @SerialVersionUID()
  15. trait ConvertableFrom [A] extends Any
  16. trait ConvertableTo [A] extends Any
  17. case class Empty [A] extends Interval[A] with Product with Serializable
  18. final class FloatComplex extends AnyVal

    Value class which encodes two floating point values in a Long.

    Value class which encodes two floating point values in a Long.

    We get (basically) unboxed complex numbers using this hack. The underlying implementation lives in the FastComplex object.

  19. final class FpFilter [A] extends AnyRef

    A Floating-point Filter [1] provides a Numeric type that wraps another Numeric type, but defers its computation, instead providing a floating point (Double) approximation.

    A Floating-point Filter [1] provides a Numeric type that wraps another Numeric type, but defers its computation, instead providing a floating point (Double) approximation. For some operations, like signum, comparisons, equality checks, toFloat, etc, the Double approximation may be used to compute the result, rather than having to compute the exact value.

    An FpFilter can generally be used with any Ring numeric type (also supports EuclideanRing, Field, and NRoot). However, it should be kept in mind that FpFilter knows nothing about the type its wrapping and assumes that, generally, it is more accurate than it is. When an FpFilter cannot determine an answer to some predicate exactly, it will defer to the wrapped value, so it probably doesn't make sense to wrap Ints, when an Int will overflow before a Double!

    Good candidates to wrap in FpFilter are BigInts, Rationals, BigDecimals, and Algebraic. Note that while Algebraic has an internal floating-point filter, this still provides benefits. Namely, the operator-fusion and allocation removal provided by the macros can make for much faster hot paths.

    Note: Both equals and hashCode will generally force the exact computation. They should be avoided (prefer === for equals)... otherwise why use bother?

    [1] Burnikel, Funke, Seel. Exact Geometric Computation Using Cascading. SoCG 1998.

  20. final class FpFilterApprox [A] extends AnyVal
  21. final class FpFilterExact [A] extends AnyVal
  22. trait Fractional [A] extends Field[A] with NRoot[A] with Integral[A]
  23. trait HighBranchingMedianOf5 extends AnyRef
  24. trait Integral [A] extends EuclideanRing[A] with ConvertableFrom[A] with ConvertableTo[A] with IsReal[A]
  25. class IntegralOps [A] extends AnyRef
  26. sealed abstract class Interval [A] extends Serializable

    Interval represents a set of values, usually numbers.

    Interval represents a set of values, usually numbers.

    Intervals have upper and lower bounds. Each bound can be one of four kinds:

    * Closed: The boundary value is included in the interval. * Open: The boundary value is excluded from the interval. * Unbound: There is no boundary value. * Empty: The interval itself is empty.

    When the underlying type of the interval supports it, intervals may be used in arithmetic. There are several possible interpretations of interval arithmetic: the interval can represent uncertainty about a single value (for instance, a quantity +/- tolerance in engineering) or it can represent all values in the interval simultaneously. In this implementation we have chosen to use the probabillistic interpretation.

    One common pitfall with interval arithmetic is that many familiar algebraic relations do not hold. For instance, given two intervals a and b:

    a == b does not imply a * a == a * b

    Consider a = b = [-1, 1]. Since any number times itself is non-negative, a * a = [0, 1]. However, a * b = [-1, 1], since we may actually have a=1 and b=-1.

    These situations will result in loss of precision (in the form of wider intervals). The result is not wrong per se, but less acccurate than it could be.

    These intervals should not be used with floating point bounds, as proper rounding is not implemented. Generally, the JVM is not an easy platform to perform robust arithmetic, as the IEEE 754 rounding modes cannot be set.

  27. final case class Jet [T](real: T, infinitesimal: Array[T]) extends ScalaNumber with ScalaNumericConversions with Serializable with Product
    Annotations
    @SerialVersionUID()
  28. case class JetDim (dimension: Int) extends Product with Serializable

    Used to implicitly define the dimensionality of the Jet space.

    Used to implicitly define the dimensionality of the Jet space.

    dimension

    the number of dimensions.

  29. trait JetInstances extends AnyRef
  30. trait Merge extends Any

    Interface for a merging strategy object.

  31. trait MutatingMedianOf5 extends AnyRef
  32. sealed abstract class Natural extends ScalaNumber with ScalaNumericConversions with Serializable
    Annotations
    @SerialVersionUID()
  33. class NaturalAlgebra extends NaturalIsRig with NaturalIsReal with Serializable
    Annotations
    @SerialVersionUID()
  34. trait NaturalInstances extends AnyRef
  35. sealed trait Number extends ScalaNumber with ScalaNumericConversions with Serializable
  36. class NumberAlgebra extends NumberIsField with NumberIsNRoot with NumberIsTrig with NumberIsReal with Serializable
    Annotations
    @SerialVersionUID()
  37. trait NumberInstances extends AnyRef
  38. trait NumberTag [A] extends AnyRef

    A NumberTag provides information about important implementations details of numbers.

    A NumberTag provides information about important implementations details of numbers. For instance, it includes information about whether we can expect arithmetic to overflow or produce invalid values, the bounds of the number if they exist, whether it is an approximate or exact number type, etc.

  39. trait Numeric [A] extends Ring[A] with AdditiveAbGroup[A] with MultiplicativeAbGroup[A] with NRoot[A] with ConvertableFrom[A] with ConvertableTo[A] with IsReal[A]

    TODO 3.

    TODO 3. LiteralOps? Literal conversions? 4. Review operator symbols? 5. Support for more operators? 6. Start to worry about things like e.g. pow(BigInt, BigInt)

  40. case class Point [A] extends Interval[A] with Product with Serializable
  41. trait Polynomial [C] extends AnyRef
  42. trait PolynomialEq [C] extends Eq[Polynomial[C]]
  43. trait PolynomialEuclideanRing [C] extends PolynomialRing[C] with EuclideanRing[Polynomial[C]] with VectorSpace[Polynomial[C], C]
  44. trait PolynomialInstances extends PolynomialInstances3
  45. trait PolynomialInstances0 extends AnyRef
  46. trait PolynomialInstances1 extends PolynomialInstances0
  47. trait PolynomialInstances2 extends PolynomialInstances1
  48. trait PolynomialInstances3 extends PolynomialInstances2
  49. trait PolynomialRig [C] extends PolynomialSemiring[C] with Rig[Polynomial[C]]
  50. trait PolynomialRing [C] extends PolynomialRng[C] with Ring[Polynomial[C]]
  51. trait PolynomialRng [C] extends PolynomialSemiring[C] with RingAlgebra[Polynomial[C], C]
  52. trait PolynomialSemiring [C] extends Semiring[Polynomial[C]]
  53. final case class Quaternion [A](r: A, i: A, j: A, k: A) extends ScalaNumber with ScalaNumericConversions with Serializable with Product
  54. trait QuaternionInstances extends AnyRef
  55. sealed abstract class Rational extends ScalaNumber with ScalaNumericConversions with Ordered[Rational]
  56. class RationalAlgebra extends RationalIsField with RationalIsReal with Serializable
    Annotations
    @SerialVersionUID()
  57. trait RationalInstances extends AnyRef
  58. sealed trait Real extends ScalaNumber with ScalaNumericConversions
  59. class RealAlgebra extends RealIsFractional
    Annotations
    @SerialVersionUID()
  60. trait RealInstances extends AnyRef
  61. trait RealIsFractional extends Fractional[Real] with Order[Real] with Signed[Real] with Trig[Real]
  62. sealed abstract class SafeLong extends ScalaNumber with ScalaNumericConversions with Ordered[SafeLong]

    Provides a type to do safe long arithmetic.

    Provides a type to do safe long arithmetic. This type will never overflow, but rather convert the underlying long to a BigInteger as need and back down to a Long when possible.

  63. trait SafeLongInstances extends AnyRef
  64. trait Select extends Any
  65. trait SelectLike extends Select

    Given a function for finding approximate medians, this will create an exact median finder.

  66. trait Sort extends Any

    Interface for a sorting strategy object.

  67. final class Trilean extends AnyVal

    Implementation of three-valued logic.

    Implementation of three-valued logic.

    This type resembles Boolean, but has three values instead of two:

    • Trilean.True (equivalent to true)
    • Trilean.False (equivalent to false)
    • Trilean.Unknown

    Trilean supports the same operations that Boolean does, and as long as all values are True or False, the results will be the same. However, the truth tables have to be extended to work with unknown:

    not: -+- T|F U|U F|T

    and: |T U F -+----- T|T U F U|U U F F|F F F

    or: |T U F -+----- T|T T T U|T U U F|T U F

    Trilean is implemented as a value type, so in most cases it will only have the overhead of a single Int. However, in some situations it will be boxed.

  68. class TrileanAlgebra extends Heyting[Trilean]
  69. final class UByte extends AnyVal with ScalaNumericAnyConversions
  70. trait UByteInstances extends AnyRef
  71. final class UInt extends AnyVal
  72. trait UIntInstances extends AnyRef
  73. final class ULong extends AnyVal
  74. trait ULongInstances extends AnyRef
  75. final class UShort extends AnyVal
  76. trait UShortInstances extends AnyRef

Value Members

  1. final def IEEEremainder(x: Double, d: Double): Double
  2. final def abs[A](a: A)(implicit ev: Signed[A]): A
  3. final def abs(n: Double): Double
  4. final def abs(n: Float): Float
  5. final def abs(n: Long): Long
  6. final def abs(n: Int): Int
  7. final def abs(n: Short): Short
  8. final def abs(n: Byte): Byte

    abs

  9. final def acos[A](a: A)(implicit ev: Trig[A]): A
  10. final def asin[A](a: A)(implicit ev: Trig[A]): A
  11. final def atan[A](a: A)(implicit ev: Trig[A]): A
  12. final def atan2[A](y: A, x: A)(implicit ev: Trig[A]): A
  13. final def cbrt(x: Double): Double
  14. final def ceil[A](a: A)(implicit ev: IsReal[A]): A
  15. final def ceil(n: BigDecimal): BigDecimal
  16. final def ceil(n: Double): Double
  17. final def ceil(n: Float): Float

    ceil

  18. def choose(n: Long, k: Long): BigInt

    choose (binomial coefficient)

  19. final def copySign(m: Float, s: Float): Float
  20. final def copySign(m: Double, s: Double): Double
  21. final def cos[A](a: A)(implicit ev: Trig[A]): A
  22. final def cosh(x: Double): Double
  23. final def cosh[A](x: A)(implicit ev: Trig[A]): A
  24. final def e[A](implicit ev: Trig[A]): A
  25. final def e: Double

    e

  26. final def exp[A](a: A)(implicit t: Trig[A]): A
  27. final def exp(k: BigDecimal): BigDecimal
  28. final def exp(k: Int, precision: Int): BigDecimal
  29. final def exp(n: Double): Double

    exp

  30. final def expm1(x: Double): Double
  31. def fact(n: Long): BigInt

    factorial

  32. def fib(n: Long): BigInt

    fibonacci

  33. final def floor[A](a: A)(implicit ev: IsReal[A]): A
  34. final def floor(n: BigDecimal): BigDecimal
  35. final def floor(n: Double): Double
  36. final def floor(n: Float): Float

    floor

  37. final def gcd[A](x: A, y: A, z: A, rest: A*)(implicit ev: EuclideanRing[A]): A
  38. final def gcd[A](xs: Seq[A])(implicit ev: EuclideanRing[A]): A
  39. final def gcd[A](x: A, y: A)(implicit ev: EuclideanRing[A]): A
  40. final def gcd(a: BigInt, b: BigInt): BigInt
  41. final def gcd(_x: Long, _y: Long): Long

    gcd

  42. final def getExponent(x: Float): Int
  43. final def getExponent(x: Double): Int
  44. final def hypot[A](x: A, y: A)(implicit f: Field[A], n: NRoot[A], o: Order[A]): A
  45. final def lcm[A](x: A, y: A)(implicit ev: EuclideanRing[A]): A
  46. final def lcm(a: BigInt, b: BigInt): BigInt
  47. final def lcm(x: Long, y: Long): Long

    lcm

  48. final def log[A](a: A, base: Int)(implicit f: Field[A], t: Trig[A]): A
  49. final def log[A](a: A)(implicit t: Trig[A]): A
  50. def log(n: BigDecimal, base: Int): BigDecimal
  51. final def log(n: BigDecimal): BigDecimal
  52. final def log(n: Double, base: Int): Double
  53. final def log(n: Double): Double

    log

  54. final def log10(x: Double): Double
  55. final def log1p(x: Double): Double
  56. final def max[A](x: A, y: A)(implicit ev: Order[A]): A
  57. final def max(x: Double, y: Double): Double
  58. final def max(x: Float, y: Float): Float
  59. final def max(x: Long, y: Long): Long
  60. final def max(x: Int, y: Int): Int
  61. final def max(x: Short, y: Short): Short
  62. final def max(x: Byte, y: Byte): Byte

    max

  63. final def min[A](x: A, y: A)(implicit ev: Order[A]): A
  64. final def min(x: Double, y: Double): Double
  65. final def min(x: Float, y: Float): Float
  66. final def min(x: Long, y: Long): Long
  67. final def min(x: Int, y: Int): Int
  68. final def min(x: Short, y: Short): Short
  69. final def min(x: Byte, y: Byte): Byte

    min

  70. final def nextAfter(x: Float, y: Float): Float
  71. final def nextAfter(x: Double, y: Double): Double
  72. final def nextUp(x: Float): Float
  73. final def nextUp(x: Double): Double
  74. final def pi[A](implicit ev: Trig[A]): A
  75. final def pi: Double

    pi

  76. final def pow(base: Double, exponent: Double): Double
  77. final def pow(base: Long, exponent: Long): Long

    Exponentiation function, e.g.

    Exponentiation function, e.g. x^y

    If base^ex doesn't fit in a Long, the result will overflow (unlike Math.pow which will return +/- Infinity).

  78. final def pow(base: BigInt, ex: BigInt): BigInt
  79. final def pow(base: BigDecimal, exponent: BigDecimal): BigDecimal

    pow

  80. final def random(): Double
  81. final def rint(x: Double): Double
  82. final def round[A](a: A)(implicit ev: IsReal[A]): A
  83. final def round(a: BigDecimal): BigDecimal
  84. final def round(a: Double): Double
  85. final def round(a: Float): Float

    round

  86. final def scalb(d: Float, s: Int): Float
  87. final def scalb(d: Double, s: Int): Double
  88. final def signum[A](a: A)(implicit ev: Signed[A]): Int
  89. final def signum(x: Float): Float
  90. final def signum(x: Double): Double

    signum

  91. final def sin[A](a: A)(implicit ev: Trig[A]): A
  92. final def sinh[A](x: A)(implicit ev: Trig[A]): A
  93. final def sqrt[A](a: A)(implicit ev: NRoot[A]): A
  94. final def sqrt(x: Double): Double

    sqrt

  95. final def tan[A](a: A)(implicit ev: Trig[A]): A
  96. final def tanh[A](x: A)(implicit ev: Trig[A]): A
  97. final def toDegrees(a: Double): Double
  98. final def toRadians(a: Double): Double
  99. final def ulp(x: Float): Double
  100. final def ulp(x: Double): Double
  101. object Algebraic extends AlgebraicInstances with Serializable
  102. object BinaryMerge extends Merge

    Merge that uses binary search to reduce the number of comparisons

    Merge that uses binary search to reduce the number of comparisons

    This can be orders of magnitude quicker than a linear merge for types that have a relatively expensive comparison operation (e.g. Rational, BigInt, tuples) and will not be much slower than linear merge even in the worst case for types that have a very fast comparison (e.g. Int)

  103. object BitString
  104. object Complex extends ComplexInstances with Serializable
  105. object ConvertableFrom
  106. object ConvertableTo
  107. object FastComplex

    FastComplex is an ugly, beautiful hack.

    FastComplex is an ugly, beautiful hack.

    The basic idea is to encode two 32-bit Floats into a single 64-bit Long. The lower-32 bits are the "real" Float and the upper-32 are the "imaginary" Float.

    Since we're overloading the meaning of Long, all the operations have to be defined on the FastComplex object, meaning the syntax for using this is a bit ugly. To add to the ugly beauty of the whole thing I could imagine defining implicit operators on Long like +@, -@, *@, /@, etc.

    You might wonder why it's even worth doing this. The answer is that when you need to allocate an array of e.g. 10-20 million complex numbers, the GC overhead of using *any* object is HUGE. Since we can't build our own "pass-by-value" types on the JVM we are stuck doing an encoding like this.

    Here are some profiling numbers for summing an array of complex numbers, timed against a concrete case class implementation using Float (in ms):

    size | encoded | class 1M | 5.1 | 5.8 5M | 28.5 | 91.7 10M | 67.7 | 828.1 20M | 228.0 | 2687.0

    Not bad, eh?

  108. object FloatComplex
  109. object FpFilter
  110. object FpFilterApprox
  111. object FpFilterExact
  112. object Fractional
  113. object InsertionSort extends Sort

    Simple implementation of insertion sort.

    Simple implementation of insertion sort.

    Works for small arrays but due to O(n^2) complexity is not generally good.

  114. object Integral
  115. object Interval extends Serializable
  116. object Jet extends JetInstances with Serializable

    A simple implementation of N-dimensional dual numbers, for automatically computing exact derivatives of functions.

    Overview

    A simple implementation of N-dimensional dual numbers, for automatically computing exact derivatives of functions. This code (and documentation) closely follow the one in Google's "Ceres" library of non-linear least-squares solvers (see Sameer Agarwal, Keir Mierle, and others: Ceres Solver.)

    While a complete treatment of the mechanics of automatic differentiation is beyond the scope of this header (see http://en.wikipedia.org/wiki/Automatic_differentiation for details), the basic idea is to extend normal arithmetic with an extra element "h" such that h != 0, but h2 = 0. Dual numbers are extensions of the real numbers analogous to complex numbers: whereas complex numbers augment the reals by introducing an imaginary unit i such that i2 = -1, dual numbers introduce an "infinitesimal" unit h such that h2 = 0. Analogously to a complex number c = x + y*i, a dual number d = x * y*h has two components: the "real" component x, and an "infinitesimal" component y. Surprisingly, this leads to a convenient method for computing exact derivatives without needing to manipulate complicated symbolic expressions.

    For example, consider the function

    f(x) = x * x ,

    evaluated at 10. Using normal arithmetic, f(10) = 100, and df/dx(10) = 20. Next, augment 10 with an infinitesimal h to get:

    f(10 + h) = (10 + h) * (10 + h)
          = 100 + 2 * 10 * h + h * h
          = 100 + 20 * h       +---
                  +-----       |
                  |            +--- This is zero
                  |
                  +----------------- This is df/dx

    Note that the derivative of f with respect to x is simply the infinitesimal component of the value of f(x + h). So, in order to take the derivative of any function, it is only necessary to replace the numeric "object" used in the function with one extended with infinitesimals. The class Jet, defined in this header, is one such example of this, where substitution is done with generics.

    To handle derivatives of functions taking multiple arguments, different infinitesimals are used, one for each variable to take the derivative of. For example, consider a scalar function of two scalar parameters x and y:

    f(x, y) = x * x + x * y

    Following the technique above, to compute the derivatives df/dx and df/dy for f(1, 3) involves doing two evaluations of f, the first time replacing x with x + h, the second time replacing y with y + h.

    For df/dx:

    f(1 + h, y) = (1 + h) * (1 + h) + (1 + h) * 3
            = 1 + 2 * h + 3 + 3 * h
            = 4 + 5 * h

Therefore df/dx = 5

    For df/dy:

    f(1, 3 + h) = 1 * 1 + 1 * (3 + h)
            = 1 + 3 + h
            = 4 + h

Therefore df/dy = 1

    To take the gradient of f with the implementation of dual numbers ("jets") in this file, it is necessary to create a single jet type which has components for the derivative in x and y, and pass them to a routine computing function f. It is convenient to use a generic version of f, that can be called also with non-jet numbers for standard evaluation:

    def f[@specialized(Double) T : Field](x: T, y: T): T = x * x + x * y

val xValue = 9.47892774
val yValue = 0.287740

// The "2" means there should be 2 dual number components.
implicit val dimension = JetDim(2)
val x: Jet[Double] = xValue + Jet.h[Double](0);  // Pick the 0th dual number for x.
val y: Jet[Double] = yValue + Jet.h[Double](1);  // Pick the 1th dual number for y.

val z: Jet[Double] = f(x, y);
println("df/dx = " + z.infinitesimal(0) + ", df/dy = " + z.infinitesimal(1));

    For the more mathematically inclined, this file implements first-order "jets". A 1st order jet is an element of the ring

    T[N] = T[t_1, ..., t_N] / (t_1, ..., t_N)^2

    which essentially means that each jet consists of a "scalar" value 'a' from T and a 1st order perturbation vector 'v' of length N:

    x = a + \sum_i v[i] t_i

    A shorthand is to write an element as x = a + u, where u is the perturbation. Then, the main point about the arithmetic of jets is that the product of perturbations is zero:

    (a + u) * (b + v) = ab + av + bu + uv
                  = ab + (av + bu) + 0

    which is what operator* implements below. Addition is simpler:

    (a + u) + (b + v) = (a + b) + (u + v).

    The only remaining question is how to evaluate the function of a jet, for which we use the chain rule:

    f(a + u) = f(a) + f'(a) u

    where f'(a) is the (scalar) derivative of f at a.

    By pushing these things through generics, we can write routines that at same time evaluate mathematical functions and compute their derivatives through automatic differentiation.

  117. object LinearMerge extends Merge

    Simple linear merge

  118. object LinearSelect extends SelectLike with HighBranchingMedianOf5
  119. object MergeSort extends Sort

    In-place merge sort implementation.

    In-place merge sort implementation. This sort is stable but does mutate the given array. It is an in-place sort but it does allocate a temporary array of the same size as the input. It uses InsertionSort for sorting very small arrays.

  120. object Natural extends NaturalInstances with Serializable
  121. object Number extends NumberInstances with Serializable

    Convenient apply and implicits for Numbers

  122. object NumberTag
  123. object Numeric
  124. object Polynomial extends PolynomialInstances

    Polynomial A univariate polynomial class and EuclideanRing extension trait for arithmetic operations.

    Polynomial A univariate polynomial class and EuclideanRing extension trait for arithmetic operations. Polynomials can be instantiated using any type C for which a Ring[C] and Signed[C] are in scope, with exponents given by Int values. Some operations require a Field[C] to be in scope.

  125. object Quaternion extends QuaternionInstances with Serializable
  126. object QuickSelect extends SelectLike with HighBranchingMedianOf5
  127. object QuickSort

    In-place quicksort implementation.

    In-place quicksort implementation. It is not stable, but does not allocate extra space (other than stack). Like MergeSort, it uses InsertionSort for sorting very small arrays.

  128. object Rational extends RationalInstances with Serializable
  129. object Real extends RealInstances with Serializable
  130. object SafeLong extends SafeLongInstances with Serializable
  131. object Searching
  132. object Selection
  133. object Sorting

    Object providing in-place sorting capability for arrays.

    Object providing in-place sorting capability for arrays.

    Sorting.sort() uses quickSort() by default (in-place, not stable, generally fastest but might hit bad cases where it's O(n^2)). Also provides mergeSort() (in-place, stable, uses extra memory, still pretty fast) and insertionSort(), which is slow except for small arrays.

  134. object Trilean
  135. object UByte extends UByteInstances
  136. object UInt extends UIntInstances
  137. object ULong extends ULongInstances
  138. object UShort extends UShortInstances

Inherited from AnyRef

Inherited from Any

Ungrouped