spire.math

Algebraic

final class Algebraic extends ScalaNumber with ScalaNumericConversions with Serializable

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:

Annotations
@SerialVersionUID( 1L )
Linear Supertypes
Serializable, ScalaNumericConversions, ScalaNumericAnyConversions, ScalaNumber, java.lang.Number, Serializable, AnyRef, Any
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  1. Algebraic
  2. Serializable
  3. ScalaNumericConversions
  4. ScalaNumericAnyConversions
  5. ScalaNumber
  6. Number
  7. Serializable
  8. AnyRef
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  1. final def !=(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  2. final def !=(arg0: Any): Boolean

    Definition Classes
    Any

  3. final def ##(): Int

    Definition Classes
    AnyRef → Any

  4. def *(that: Algebraic): Algebraic

  5. def +(that: Algebraic): Algebraic

  6. def -(that: Algebraic): Algebraic

  7. def /(that: Algebraic): Algebraic

  8. def <(that: Algebraic): Boolean

  9. def <=(that: Algebraic): Boolean

  10. def =!=(that: Algebraic): Boolean

  11. final def ==(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  12. final def ==(arg0: Any): Boolean

    Definition Classes
    Any

  13. def ===(that: Algebraic): Boolean

  14. def >(that: Algebraic): Boolean

  15. def >=(that: Algebraic): Boolean

  16. def abs: Algebraic

    Return a non-negative Algebraic with the same magnitude as this one.

  17. final def asInstanceOf[T0]: T0

    Definition Classes
    Any

  18. def byteValue(): Byte

    Definition Classes
    Number

  19. def cbrt: Algebraic

    Returns the cube root of this number.

  20. def clone(): AnyRef

    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  21. def compare(that: Algebraic): Int

    Returns an integer with the same sign as this - that.

    Returns an integer with the same sign as this - that. Specifically, if this < that, then the sign is negative, if this > that, then the sign is positive, otherwise this == that and this returns 0.

  22. def doubleValue(): Double

    Returns a Double that approximates this value.

    Returns a Double that approximates this value. If the exponent is too large to fit in a double, the Double.PositiveInfinity or Double.NegativeInfinity is returned.

    Definition Classes
    Algebraic → ScalaNumericAnyConversions → Number

  23. final def eq(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  24. def equals(that: Any): Boolean

    Definition Classes
    Algebraic → AnyRef → Any

  25. def evaluateWith[A](implicit arg0: Field[A], arg1: NRoot[A], arg2: RootFinder[A], arg3: Eq[A], arg4: ClassTag[A], conv: ConvertableTo[A]): A

    Evaluates this algebraic expression with a different number type.

    Evaluates this algebraic expression with a different number type. All Algebraic numbers store the entire expression tree, so we can use this to *replay* the stored expression using a different type. This will accumulate errors as if the number type had been used from the beginning and is only really suitable for more exact number types, like Real.

    TODO: Eq/ClassTag come from poly.map - would love to get rid of them.

  26. val expr: Expr

  27. def finalize(): Unit

    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] )

  28. def floatValue(): Float

    Returns a Float that approximates this value.

    Returns a Float that approximates this value. If the exponent is too large to fit in a float, the Float.PositiveInfinity or Float.NegativeInfinity is returned.

    Definition Classes
    Algebraic → ScalaNumericAnyConversions → Number

  29. final def getClass(): Class[_]

    Definition Classes
    AnyRef → Any

  30. def hashCode(): Int

    Definition Classes
    Algebraic → AnyRef → Any

  31. def intValue(): Int

    Returns the nearest, valid Int value to this Algebraic, without going further away from 0 (eg.

    Returns the nearest, valid Int value to this Algebraic, without going further away from 0 (eg. truncation).

    If this Algebraic represented 1.2, then this would return 1. If this represented -3.3, then this would return -3. If this value is greater than Int.MaxValue, then Int.MaxValue is returned. If this value is less than Int.MinValue, then Int.MinValue is returned.

    Definition Classes
    Algebraic → ScalaNumericAnyConversions → Number

  32. final def isInstanceOf[T0]: Boolean

    Definition Classes
    Any

  33. def isRational: Boolean

    Returns true iff this is a rational expression (ie contains no n-root expressions).

    Returns true iff this is a rational expression (ie contains no n-root expressions). Otherwise it is a radical expression and returns false.

  34. def isValidByte: Boolean

    Definition Classes
    ScalaNumericAnyConversions

  35. def isValidChar: Boolean

    Definition Classes
    ScalaNumericAnyConversions

  36. def isValidInt: Boolean

    Returns true if this Algebraic number is a whole number (no fractional part) and fits within the bounds of an Int.

    Returns true if this Algebraic number is a whole number (no fractional part) and fits within the bounds of an Int. That is, if x.isValidInt, then Algebraic(x.toInt) == x.

    Definition Classes
    Algebraic → ScalaNumericAnyConversions

  37. def isValidLong: Boolean

    Returns true if this Algebraic number is a whole number (no fractional part) and fits within the bounds of an Long.

    Returns true if this Algebraic number is a whole number (no fractional part) and fits within the bounds of an Long. That is, if x.isValidLong, then Algebraic(x.toLong) == x.

  38. def isValidShort: Boolean

    Definition Classes
    ScalaNumericAnyConversions

  39. def isWhole(): Boolean

    Returns true iff this Algebraic exactly represents a valid BigInt.

    Returns true iff this Algebraic exactly represents a valid BigInt.

    Definition Classes
    Algebraic → ScalaNumericAnyConversions → ScalaNumber

  40. def isZero: Boolean

    Returns true iff this Algebraic number is exactly 0.

  41. def longValue(): Long

    Returns the nearest, valid Long value to this Algebraic, without going further away from 0 (eg.

    Returns the nearest, valid Long value to this Algebraic, without going further away from 0 (eg. truncation).

    If this Algebraic represented 1.2, then this would return 1. If this represented -3.3, then this would return -3. If this value is greater than Long.MaxValue, then Long.MaxValue is returned. If this value is less than Long.MinValue, then Long.MinValue is returned.

    Definition Classes
    Algebraic → ScalaNumericAnyConversions → Number

  42. final def ne(arg0: AnyRef): Boolean

    Definition Classes
    AnyRef

  43. final def notify(): Unit

    Definition Classes
    AnyRef

  44. final def notifyAll(): Unit

    Definition Classes
    AnyRef

  45. def nroot(k: Int): Algebraic

    Returns the k-th root of this number.

  46. def pow(k: Int): Algebraic

    Raise this number to the k-th power.

  47. def shortValue(): Short

    Definition Classes
    Number

  48. def sign: Sign

    Returns the sign of this Algebraic number.

    Returns the sign of this Algebraic number. Algebraic numbers support exact sign tests, so this is guaranteed to be accurate.

  49. def signum: Int

    Returns an Int with the same sign as this algebraic number.

    Returns an Int with the same sign as this algebraic number. Algebraic numbers support exact sign tests, so this is guaranteed to be accurate.

  50. def sqrt: Algebraic

    Returns the square root of this number.

  51. final def synchronized[T0](arg0: ⇒ T0): T0

    Definition Classes
    AnyRef

  52. def toBigDecimal(mc: MathContext): BigDecimal

    Relative approximation to the precision specified in mc with the given rounding mode.

    Relative approximation to the precision specified in mc with the given rounding mode. Rounding is always exact. The sign is always correct; the sign of the returned BigDecimal matches the sign of the exact value this Algebraic represents.

    mc

    the precision and rounding mode of the final result

    returns

    an approximation to the value of this algebraic number

  53. def toBigDecimal(scale: Int, roundingMode: RoundingMode): BigDecimal

    Absolute approximation to scale decimal places with the given rounding mode.

    Absolute approximation to scale decimal places with the given rounding mode. Rounding is always exact.

  54. def toBigInt: BigInt

    Returns the nearest, valid BigInt value to this Algebraic, without going further away from 0 (eg.

    Returns the nearest, valid BigInt value to this Algebraic, without going further away from 0 (eg. truncation).

    If this Algebraic represented 1.2, then this would return 1. If this represented -3.3, then this would return -3.

  55. def toByte: Byte

    Definition Classes
    ScalaNumericAnyConversions

  56. def toChar: Char

    Definition Classes
    ScalaNumericAnyConversions

  57. def toDouble: Double

    Definition Classes
    ScalaNumericAnyConversions

  58. def toExprString: String

  59. def toFloat: Float

    Definition Classes
    ScalaNumericAnyConversions

  60. def toInt: Int

    Definition Classes
    ScalaNumericAnyConversions

  61. def toLong: Long

    Definition Classes
    ScalaNumericAnyConversions

  62. def toRational: Option[Rational]

    If this is a rational expressions, then it returns the exact value as a Rational.

    If this is a rational expressions, then it returns the exact value as a Rational. Otherwise, this is a radical expression and None is returned.

  63. def toReal: Real

    Returns an exact Real representation of this number.

  64. def toShort: Short

    Definition Classes
    ScalaNumericAnyConversions

  65. def toString(): String

    Definition Classes
    Algebraic → AnyRef → Any

  66. def unary_-: Algebraic

  67. def underlying(): AnyRef

    Definition Classes
    Algebraic → ScalaNumericConversions → ScalaNumericAnyConversions → ScalaNumber

  68. def unifiedPrimitiveEquals(x: Any): Boolean

    Attributes
    protected
    Definition Classes
    ScalaNumericAnyConversions

  69. def unifiedPrimitiveHashcode(): Int

    Attributes
    protected
    Definition Classes
    ScalaNumericAnyConversions

  70. final def wait(): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  71. final def wait(arg0: Long, arg1: Int): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

  72. final def wait(arg0: Long): Unit

    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

Inherited from Serializable

Inherited from ScalaNumericConversions

Inherited from ScalaNumericAnyConversions

Inherited from ScalaNumber

Inherited from java.lang.Number

Inherited from Serializable

Inherited from AnyRef

Inherited from Any

Ungrouped