Category talk:Wren-complex

Complex numbers

This module aims to provide broadly similar functionality for complex numbers as already exists for real numbers. However, as the former - taken as a whole - are unordered, methods which require ordering have been omitted.

Complex numbers can be constructed from two separate Nums, from a pair of Nums, from a Rat object, from a complex string or (by copying) from another Complex number . These representations can also be mixed in operations as long as the first operand is itself a Complex object.

Support for complex matrices is also included though is not as extensive as for real matrices in the Wren-matrix module. It does however include some extra methods (such as conjugate transpose) which only apply to complex matrices.

Source code

<lang ecmascript>/* Module "complex.wren" */

import "/trait" for Cloneable

/* Complex represents a complex number of the form 'a + bi' where 'a' and 'b'

  are both Nums. Complex objects are immutable.

• /

class Complex is Cloneable {

   // Private helper function to check that 'o' is a suitable type and throw an error
   // otherwise. Real numbers, rationals and numeric strings are returned as complex numbers.
   static check_(o) {
       if (o is Complex) return o
       if (o is Num) return new_(o, 0)
       if ((o is List) && o.count == 2 && (o[0] is Num) && (o[1] is Num)) {
           return Complex.new_(o[0], o[1])
       }
       if (o.type.toString == "Rat") return new_(r.toFloat, 0)
       if (o is String) return fromString(o)
       Fiber.abort("Argument must be a number, pair of numbers, a rational number or a complex string.")
   }

   // Constants.
   static minusOne     { Complex.new_(-1,  0) }
   static zero         { Complex.new_( 0,  0) }
   static one          { Complex.new_( 1,  0) }
   static two          { Complex.new_( 2,  0) }
   static ten          { Complex.new_(10,  0) } 
   static imagMinusOne { Complex.new_( 0, -1) }
   static imagOne      { Complex.new_( 0,  1) }
   static imagTwo      { Complex.new_( 0,  2) }
   static imagTen      { Complex.new_( 0, 10) }
   static i            { Complex.new_( 0,  1) } // same as imagOne

   static pi           { Complex.new_(Num.pi, 0) }
   static e            { Complex.new_(2.71828182845904523536, 0) }
   static phi          { Complex.new_(1.6180339887498948482,  0) } // golden ratio
   static tau          { Complex.new_(1.6180339887498948482,  0) } // synonym for phi
   static ln2          { Complex.new_(0.69314718055994530942, 0) } // 2.log
   static ln10         { Complex.new_(2.30258509299404568402, 0) } // 10.log

   // Determines whether a Complex object is always shown as such
   // or, if purely real, as a real.
   static showAsReal     { __showAsReal }
   static showAsReal=(b) { __showAsReal = b }

   // Constructs a new Complex object by passing it real and imaginary components.
   construct new(real, imag) {
       if (real.type != Num || imag.type != Num) System.print("Argument(s) must be numbers.")
       _real = real
       _imag = imag
   }

   // Convenience method which constructs a new Complex object from a Num by passing it
   // just a real component.
   static new(real) { new(real, 0) }

   // Private constructor which avoids type checking.
   construct new_(real, imag) {
       _real = real
       _imag = imag
   }

   // Constructs a Complex object from an ordered pair of numbers [real, imag].
   static fromPair(p) {
      if (p.type != List || p.count != 2 || p[0].type != Num || p[1].type != Num) {
           Fiber.abort("Argument must be an ordered pair of numbers.")
      }
      return Complex.new_(p[0], p[1])
   }

   // Constructs a Complex object from a string of the form '±a±bi', '±a' or '±bi'
   // where 'a' and 'b' are string representations of Nums.
   static fromString(s) {
       if (s.type != String) Fiber.abort("Argument must be a complex string.")
       s = s.trim()
       if (s == "") Fiber.abort("Invalid complex string.")
       s = s.replace("--", "+")
       if (s[0] == "+") s = s[1..-1]
       if (s == "") Fiber.abort("Invalid complex string.")
       s = s.replace("e+", "e")
       var neg = s[0] == "-"
       if (neg) {
           s = s[1..-1]
           if (s == "") Fiber.abort("Invalid complex string.")
       }
       var negExp = s.indexOf("e-") >= 0
       if (negExp) s = s.replace("e-", "\v")
       if (s.indexOf("+") >= 0) {
           var split = s.split("+")
           if (split.count != 2) Fiber.abort("Invalid complex string.")
           if (negExp) for (i in 0..1) split[i] = split[i].replace("\v", "e-")
           var real = Num.fromString(split[0])
           if (!real) Fiber.abort("Invalid real part.")
           if (neg) real = -real
           if (!split[1].endsWith("i")) Fiber.abort("Invalid complex string.")
           var imag = Num.fromString(split[1][0..-2])
           if (!imag) Fiber.abort("Invalid imaginary part.")
           return Complex.new_(real, imag)
       } else if (s.indexOf("-") >= 0) {
           var split = s.split("-")
           if (split.count != 2) Fiber.abort("Invalid complex string.")
           if (negExp) for (i in 0..1) split[i] = split[i].replace("\v", "e-")
           var real = Num.fromString(split[0])
           if (!real) Fiber.abort("Invalid real part.")
           if (neg) real = -real
           if (!split[1].endsWith("i")) Fiber.abort("Invalid complex string.")
           var imag = Num.fromString(split[1][0..-2])
           if (!imag) Fiber.abort("Invalid imaginary part.")
           return Complex.new_(real, -imag)
       } else if (s.endsWith("i")) {
           if (negExp) s = s.replace("\v", "e-")
           var imag = Num.fromString(s[0..-2])
           if (!imag) Fiber.abort("Invalid imaginary part.")
           if (neg) imag = -imag
           return Complex.new_(0, imag)
       } else {
           if (negExp) s = s.replace("\v", "e-")
           var real = Num.fromString(s)
           if (!real) Fiber.abort("Invalid real part.")
           if (neg) real = -real
           return Complex.new_(real, 0)
       }
   }

   // Constructs a Complex object from a Rat object.
   static fromRat(r) {
       if (r.type.toString != "Rat") Fiber.abort("Argument must be a rational number.")
       return Complex.new_(r.toFloat, 0)
   }

   // Constructs a Complex object from polar coordinates (r, theta).
   static fromPolar(r, theta)  {
       if (r.type != Num || theta.type != Num) {
           Fiber.abort("Arguments must be numbers.")      
       }
       return Complex.new_(r * theta.cos, r * theta.sin)
   }

   // Basic properties.
   real { _real }  // real component
   imag { _imag }  // imaginary component

   isInfinity { _real.isInfinity || _imag.isInfinty } // true if either part is infinite
   isNan      { _real.isNan || imag.isNan }           // true if either part is nan

   // Returns whether real component is an integer and imaginary component is zero.
   isRealInteger { _real.isInteger && _imag == 0 }

   // Returns whether imaginary component is an integer and real component is zero.
   isImagInteger { _imag.isInteger && _real == 0 }

   // Returns the ordered pair [_real, _imag].
   toPair { [_real, _imag] }

   // Returns the polar coordinates of this instance [modulus, phase].
   toPolar { [abs, phase] }

   // Returns a new instance which negates the current one.
   - { Complex.new_(-_real, -_imag) }

   // Returns the inverse or reciprocal of this instance.
   inverse {
       var denom = _real * _real + _imag * _imag
       return Complex.new_(_real/denom, -_imag/denom)
   }

   // Arithmetic operators (work with real numbers, rational numbers, complex strings
   // as well as other complex numbers). Always return a new instance.
   +(o) {
       o = Complex.check_(o)
       return Complex.new_(_real + o.real, _imag + o.imag)
   }

   -(o) { this + (-o) }

   *(o) {
       o = Complex.check_(o)
       return Complex.new_(
           _real * o.real - _imag * o.imag,
           _real * o.imag + _imag * o.real
       )
   }

   /(o) { 
       o = Complex.check_(o)
       var i = o.inverse
       return Complex.new_(
           _real * i.real - _imag * i.imag,
           _real * i.imag + _imag * i.real
       )
   }

   // Returns the absolute value or modulus of this instance.
   abs { (_real*_real + _imag*_imag).sqrt }

   // Returns the phase or argument of the current instance in the range [-π, π].
   phase { _imag.atan(_real) }

   // Returns the complex conjugate of this instance
   conj { Complex.new_(_real, -_imag) }

   // Returns the square of this instance.
   square { Complex.new_(_real * _real - _imag * _imag, _real * _imag * 2) }

   // Returns the square root of this instance.
   sqrt {
       var m = abs
       var r = ((m + _real)/2).sqrt
       var i = ((m - _real)/2).sqrt
       if (_imag < 0) i = -i
       return Complex.new_(r, i)
   }

   // Returns the base 'e' exponential of this instance.
   exp {
       var e = Complex.e.real.pow(_real) /* change to _real.exp from version 0.4.0 */
       return Complex.new_(e * _imag.cos, e * _imag.sin)
   }

   // Returns the natural logarithm of the current instance.
   log {
       var p = phase
       if (p > Num.pi) p = p - Num.pi*2
       return Complex.new_(abs.log, p)
   }

   // Returns the logarithm to the base 2 of the current instance.
   log2  { log / Complex.two.log }

   // Returns the logarithm to the base 10 of the current instance.
   log10 { log / Complex.ten.log }

   // Returns this instance to the power of the complex number 'e'.
   pow(e) {
       e = Complex.check_(e)
       return (log * e).exp
   }

   // Returns the cosine of the current instance.
   cos {
       var i = Complex.i
       return ((i * this).exp + (i * (-this)).exp) / Complex.two
   }

   // Returns the sine of the current instance.
   sin {
       var i = Complex.i
       return ((i * this).exp - (i * (-this)).exp) / Complex.imagTwo
   }

   // Returns the tangent of the current instance.
   tan { sin / cos }

   // Returns the arc cosine of the current instance.
   acos {
       var c = (Complex.one - square).sqrt
       c = this + c * Complex.imagMinusOne
       return c.log * Complex.i
   }

   // Returns the arc sine of the current instance.
   asin {
       var c = (Complex.one - square).sqrt
       c = c + this * Complex.imagMinusOne
       return c.log * Complex.i
   }

   // Returns the arc tangent of the current instance.
   atan {
       var a = Complex.new_(_real, _imag - 1)
       var b = Complex.new_(-_real, -_imag - 1)
       return (Complex.imagMinusOne * (a/b).log) / Complex.two
   }

   // Returns the hyperbolic cosine of the current instance.
   cosh { (this.exp + (-this).exp)/Complex.two }

   // Returns the hyperbolic sine of the current instance.
   sinh { (this.exp - (-this).exp)/Complex.two }

   // Returns the hyperbolic tangent of the current instance.
   tanh { sinh/cosh }

   // Returns the inverse hyperbolic cosine of the current instance.
   acosh { (this + (square - Complex.one).sqrt).log }

   // Returns the inverse hyperbolic sine of the current instance.
   asinh { (this + (square + Complex.one).sqrt).log }

   // Returns the inverse hyperbolic tangent of the current instance.
   atanh {
       var c = (this + Complex.one).log
       c = c - (-(this - Complex.one)).log
       return c / Complex.two
   }

   // The inherited 'clone' method just returns 'this' as Complex objects are immutable.
   // If you need an actual copy use this method instead.
   copy() { Complex.new_(_real, _imag ) }

   // Equality operators.
   ==(o) { 
       o = Complex.check_(o)
       return _real == o.real && _imag == o.imag 
   }
   !=(o) { !(this == o) }

   // Returns the string representation of this Complex object depending on 'showAsReal'.
   toString {
       if (_real == -0) _real = 0
       if (_imag == -0) _imag = 0
       var s = (_imag >= 0) ? "%(_real) + %(_imag)" : "%(_real) - %(-_imag)"
       s = (_imag.abs != 1) ? s + "i" : s[0..-2] + "i"
       if (s.endsWith("- 0i")) s = s[0..-5] + "+ 0i"
       return (Complex.showAsReal && _imag == 0) ? s = s[0..-6] : s
   }

}

/* Complexes contains routines applicable to lists of complex numbers. */ class Complexes {

   static sum(a)  { a.reduce(Complex.zero) { |acc, x| acc + x } }
   static mean(a) { sum(a)/a.count }
   static prod(a) { a.reduce(Complex.one)  { |acc, x| acc * x } }

}

/* CMatrix represents a two dimensional list of complex numbers. Once created the number of

  rows and columns of the matrix cannot be changed but individual elements can be.

• /

class CMatrix {

   // Returns an instance of the identity matrix for a given number of rows.
   static identity(numRows) {
       if (numRows.type != Num || !numRows.isInteger || numRows < 1) {
           Fiber.abort("Number of rows must be a positive integer.")
       }
       var id = new_(numRows, numRows, Complex.zero)
       for (i in 0...numRows) id.set_(i, i, Complex.one)
       return id
   }

   // Constructs a new CMatrix object by passing it the number of rows and
   // columns and the initial value for each element.
   construct new(numRows, numCols, filler) {
       if (numRows.type != Num || !numRows.isInteger || numRows < 1) {
           Fiber.abort("Number of rows must be a positive integer.")
       }
       if (numCols.type != Num || !numCols.isInteger || numCols < 1) {
           Fiber.abort("Number of columns must be a positive integer.")
       }
       if (filler.type != Complex && filler.type != Num) {
           Fiber.abort("Filler must be a complex or real number.")
       }
       if (filler.type == Num) filler = Complex.new_(filler, 0)
       _a = List.filled(numRows, null)
       for (i in 0...numRows) _a[i] = List.filled(numCols, filler)
       _nr = numRows
       _nc = numCols
   }

   // Convenience version of the public constructor which uses a filler of zero.
   static new(numRows, numCols) { new(numRows, numCols, Complex.zero) }

   // Private version of above constructor to avoid type checks.
   construct new_(numRows, numCols, filler) {
       _a = List.filled(numRows, null)
       for (i in 0...numRows) _a[i] = List.filled(numCols, filler)
       _nr = numRows
       _nc = numCols
   }

   // Constructs a new CMatrix object from a two dimensional list of complex numbers.
   construct new(a) {
       if (a.type != List || a.count == 0 || a[0].type != List || a[0].count == 0 || a[0][0].type != Complex) {
           Fiber.abort("Argument must be a non-empty two dimensional list of complex numbers.")
       }
       _nr = a.count
       _nc = a[0].count
       // copy the list so it can be mutated independently
       _a = List.filled(_nr, null)
       for (i in 0..._nr) _a[i] = a[i].toList
   }

   // Private version of above constructor to avoid type checks and copying.
   construct new_(a) {
       _a  = a
       _nr = a.count
       _nc = a[0].count
   }

   // Constructs a new CMatrix object from a two dimensional list of real numbers.
   static fromReals(a) {
       if (a.type != List || a.count == 0 || a[0].type != List || a[0].count == 0 || a[0][0].type != Num) {
           Fiber.abort("Argument must be a non-empty two dimensional list of real numbers.")
       }
       var ca = List.filled(a.count, null)
       for (i in 0...a.count) {
           ca[i] = List.filled(a[0].count, null)
           for (j in 0...a[0].count) ca[i][j] = Complex.new(a[i][j])
       }
       return new_(ca)
   }

   // Basic properties.
   numRows     { _nr }         // returns the number of rows
   numCols     { _nc }         // returns the number of columns
   size        { [_nr, _nc] }  // returns both the above in a list
   numElements { _nr * _nc  }  // returns the number of elements
   first       { _a[0][0]   }  // returns the first element
   last        { _a[-1][-1] }  // returns the last element

   // Creates another CMatrix by multiplying all elements of the current instance by -1.
   - { this * Complex.minusOne }

   // Creates another CMatrix by either:
   // 1. adding another CMatrix of the same size to the current instance; or
   // 2. adding a complex or real number to each element of the current instance.
   +(b) {
       if (b is Num) b = Complex.new_(b, 0)
       var c = List.filled(_nr, null)
       if (b is Complex) {
           for (i in 0..._nr) {
               c[i] = List.filled(_nc, null)
               for (j in 0..._nc) c[i][j] = _a[i][j] + b
           }
       } else if (b is CMatrix) {
           if (!sameSize(b)) Fiber.abort("Matrices must be of the same size.")
           for (i in 0..._nr) {
               c[i] = List.filled(_nc, null)
               for (j in 0..._nc) c[i][j] = _a[i][j] + b.get_(i, j)
           }
       } else {
           Fiber.abort("Argument must be a complex matrix, a complex number or a real number.")
       }
       return CMatrix.new_(c)
   }

   // Creates another CMatrix by either:
   // 1. subtracting another CMatrix of the same size from the current instance; or
   // 2. subtracting a complex or real number from each element of the current instance.
   -(b) { this + (-b) }

   // Creates another CMatrix by either:
   // 1. multiplying the current instance by another CMatrix of appropriate size; or
   // 2. multiplying each element of the current instance by a complex or real number.
   *(b) {
       if (b is Num) b = Complex.new_(b, 0)
       var c = List.filled(_nr, null)
       if (b is Complex) {
           for (i in 0..._nr) {
               c[i] = List.filled(_nc, null)
               for (j in 0..._nc) c[i][j] = _a[i][j] * b
           }
       } else if (b is CMatrix) {
           if (_nc != b.numRows) Fiber.abort("Cannot multiply these matrices.")
           for (i in 0..._nr) {
               c[i] = List.filled(b.numCols, Complex.zero)
               for (j in 0...b.numCols) {
                   for (k in 0..._nc) c[i][j] = c[i][j] + _a[i][k] * b.get_(k, j)
               }
           }
       } else {
           Fiber.abort("Argument must be a complex matrix, a complex number or a real number.")
       }
       return CMatrix.new_(c)
   }

   // Creates another CMatrix by dividing each element of the current instance by a complex or real number.
   /(n) {
       if (n is Num) n = Complex.new_(n, 0)
       return this * n.inverse
   }

   // Synomym for pow(n).
   ^(n) { pow(n) }

   // Creates another CMatrix by applying the 'abs' method to each element of the
   // current instance.
   abs { apply { |e| e.abs } }

   // Creates another CMatrix by multiplying the current instance by itself 'n' times.
   pow(n) {
       if (n.type != Num || !n.isInteger || n < 0) {
           Fiber.abort("Argument must be a non-negative integer.")
       }
       if (n == 0) return CMatrix.identity(_nr)
       if (n == 1) return this.copy()
       var p = CMatrix.identity(_nr)
       var base = this.copy()
       while (n > 0) {
           if ((n & 1) == 1) p = p * base
           n = n >> 1
           base = base * base
       }
       return p
   }

   // Private methods to check that a row or column number are valid.
   validRowNum_(rn) { rn.type == Num && rn.isInteger && rn >= 0 && rn < _nr }
   validColNum_(cn) { cn.type == Num && cn.isInteger && cn >= 0 && cn < _nc }

   // Returns a copy of this instance's 'i'th row.
   row(i) { validRowNum_(i) ? _a[i].toList : Fiber.abort("Invalid row number.") }

   // Returns a copy of this instance's 'i'th column.
   col(i) {
       if (!validColNum_(i)) Fiber.abort("Invalid column number.")
       var t = List.filled(_nc, null)
       for (r in 0..._nr) t[r] = _a[r][i]
       return t
   }

   // Returns a copy of this instance's main diagonal as long as its square.
   diag {
       if (!isSquare) Fiber.abort("Matrix must be square.")
       var d = List.filled(_nr, null)
       for (i in 0..._nr) d[i] = _a[i][i]
       return d
   }

   // Returns a copy of this instance's 'i'th row (synonym for row(i)).
   [i] { row(i) }

   // Returns the element at row 'i' and column 'j' of the current instance.
   [i, j] { (validRowNum_(i) && validColNum_(j)) ? _a[i][j] : Fiber.abort("Out of range.") }

   // Sets the element at row 'i' and column 'j' of the current instance to value 'v'.
   [i, j]=(v) {
       if (!validRowNum_(i) || !validColNum_(j)) Fiber.abort("Out of range.")
       if (v.type != Complex && v.type != Num) Fiber.abort("Element value must be a complex or real number.")
       if (v.type == Num) v = Complex.new_(v, 0)
       _a[i][j] = v
   }

   // Private methods to get or set the elements at row 'i' and column 'j' of the current
   // instance without any validity checks.
   get_(i, j)    { _a[i][j] }
   set_(i, j, v) { _a[i][j] = v }

   // Returns whether or not this instance is the same size as another CMatrix or Matrix.
   sameSize(b) { _nr == b.numRows && _nc == b.numCols }

   // Various self-explanatory properties.
   isSquare        { _nr == _nc }
   isRowVector     { _nr == 1 }
   isColVector     { _nc == 1 }
   isSymmetric     { isSquare && this == this.transpose }
   isSkewSymmetric { isSquare && this == -this.transpose }
   isOrthogonal    { isSquare && inverse == transpose }
   isIdempotent    { isSquare && (this * this == this) }
   isInvolutory    { isSquare && (this * this == CMatrix.identity(_nr)) }
   isSingular      { det == Complex.zero }

   isHermitian     { isSquare && this == this.conjTranspose }
   isSkewHermitian { isSquare && this == -this.conjTranspose }
   isNormal        { isSquare && this * conjTranspose == conjTranspose * this }
   isUnitary       { isSquare && this * conjTranspose == CMatrix.identity(_nr) }

   // Returns whether all the elements of the current instance outside the main diagonal
   // are zero.
   isDiagonal {
       if (!isSquare) return false
       for (i in 0..._nr) {
           for (j in 0..._nr) {
               if (i != j && _a[i][j] != Complex.zero) return false
           }
       }
       return true
   }

   // Returns whether all the current instance's elements above the main diagonal are zero.
   isLowerTriangular {
       if (!isSquare) return false
       for (i in 0..._nr - 1) {
           for (j in i + 1..._nr) {
               if (_a[i][j] != Complex.zero) return false
           }
       }
       return true
   }

   // Returns whether all the current instance's elements below the main diagonal are zero.
   isUpperTriangular {
       if (!isSquare) return false
       for (i in 1..._nr) {
           for (j in 0...i) {
               if (_a[i][j] != Complex.zero) return false
           }
       }
       return true
   }

   // Returns whether the current instance is lower or upper triangular.
   isTriangular { isLowerTrinagular || isUpperTriangular }

   // Returns the conjugate of the current instance.
   conj {
       var c = CMatrix.new_(_nr, _nc, Complex.zero)
       for (i in 0..._nc) {
           for (j in 0..._nr) c.set_(i, j, _a[i][j].conj)
       }
       return c
   }

   // Returns the transpose of the current instance.
   transpose {
       var t = CMatrix.new_(_nc, _nr, Complex.zero)
       for (i in 0..._nc) {
           for (j in 0..._nr) t.set_(i, j, _a[j][i])
       }
       return t
   }

   // Returns the conjugate transpose of the current instance.
   conjTranspose {
       var ct = CMatrix.new_(_nc, _nr, Complex.zero)
       for (i in 0..._nc) {
           for (j in 0..._nr) ct.set_(i, j, _a[j][i].conj)
       }
       return ct
   }

   // Returns a new CMatrix formed by applying a function ( Complex -> Complex )
   // to each element of the current instance.
   apply(f) {
       var t = CMatrix.new_(_nc, _nr, Complex.zero)
       for (i in 0..._nr) {
           for (j in 0..._nc) t.set_(i, j, f.call(_a[i][j]))
       }
       return t
   }

   // Transforms the current instance by applying a function ( Complex -> Complex )
   // to each of its elements.
   transform(f) {
       for (i in 0..._nr) {
           for (j in 0..._nc) _a[i][j] = f.call(_a[i][j])
       }
   }

   // Changes all elements of the current instance by multiplying them by 'm'
   // and then adding 'a'.
   changeAll(m, a) {
       if ((m.type != Complex && m != Num) || (a.type != Complex && a.type != Num)) {
           Fiber.abort("Multiplier and addend must be complex or real numbers.")
       }
       for (i in 0..._nr) {
           for (j in 0..._nc) _a[i][j] = _a[i][j]*m + a
       }
   }

   // Changes all elements of a specified row of the current instance by multiplying
   // them by 'm' and then adding 'a'.
   changeRow(rowNum, m, a) {
       if (!validRowNum_(rowNum)) Fiber.abort("Invalid row number.")
       if ((m.type != Complex && m != Num) || (a.type != Complex && a.type != Num)) {
           Fiber.abort("Multiplier and addend must be complex or real numbers.")
       }
       for (j in 0..._nc) _a[rowNum][j] = _a[rowNum][j]*m + a
   }

   // Changes all elements of a specified column of the current instance by multiplying
   // them by 'm' and then adding 'a'.
   changeCol(colNum, m, a) {
       if (!validColNum_(colNum)) Fiber.abort("Invalid column number.")
       if ((m.type != Complex && m != Num) || (a.type != Complex && a.type != Num)) {
           Fiber.abort("Multiplier and addend must be complex or real numbers.")
       }
       for (i in 0..._nr) _a[i][colNum] = _a[i][colNum]*m + a
   }

   // Swaps two specified rows of the current instance.
   swapRows(rowNum1, rowNum2) {
       if (!validRowNum_(rowNum1) || !validRowNum_(rowNum2)) Fiber.abort("Invalid row number.")
       swapRows_(rowNum1, rowNum2)
   }

   // Private method to swap two rows of the current instance without checking validity.
   swapRows_(rowNum1, rowNum2) {
       if (rowNum1 == rowNum2) return
       var t = row(rowNum1)
       for (j in 0..._nc) {
           _a[rowNum1][j] = _a[rowNum2][j]
           _a[rowNum2][j] = t[j]
       }
   }

   // Swaps two specified columns of the current instance.
   swapCols(colNum1, colNum2) {
       if (!validColNum_(colNum1) || !validColNum_(colNum2)) Fiber.abort("Invalid column number.")
       if (colNum1 == colNum2) return
       var t = col(colNum1)
       for (i in 0..._nr) {
           _a[i][colNum1] = _a[i][colNum2]
           _a[i][colNum2] = t[i]
       }
   }

   // Copies the elements of the current instance to a 2D list.
   toList {
       var l = List.filled(_nr, null)
       for (i in 0..._nr) l[i] = _a[i].toList
       return l
   }

   // Flattens the current instance by transferring all its elements row by row
   // to a new single dimensional list.
   flatten() {
       var t = []
       for (i in 0..._nr) t.addAll(_a[i])
       return t
   }

   // Returns a copy of this instance
   copy() { CMatrix.new_(this.toList) }

   // Checks whether or not the current instance's elements all have the same
   // values as the corresponding elements of another CMatrix.
   ==(b) {
       if (b.type != CMatrix) Fiber.abort("Argument must be a complex matrix.")
       if (!sameSize(b)) return false
       for (i in 0..._nr) {
           for (j in 0..._nc) if (_a[i][j] != b.get_(i, j)) return false
       }
       return true
   }

   // Checks whether or not all the current instance's elements do not have the same
   // values as the corresponding elements of another CMatrix.
   !=(b) { !(this == b) }

   // Checks whether or not the current instance's elements all have the same values
   // as the corresponding elements of another CMatrix to within a specified tolerance,
   almostEquals(b, tol) {
       if (b.type != CMatrix) Fiber.abort("Argument must be a complex matrix.")
       if (!sameSize(b)) return false
       if (tol.type != Num || tol <= 0 || tol >= 1e-5) {
           Fiber.abort("Tolerance must be a positive number <= 1e-5.")
       }
       var d = this - b
       for (i in 0..._nr) {
           for (j in 0..._nc) if (d.get_(i, j).abs > tol) return false
       }
       return true
   }

   // Convenince version of above method which uses a tolerance of 1e-14.
   almostEquals(b) { almostEquals(b, 1e-14) }

   // Returns a minor of the current instance after removing a specified row
   // and a specified column.
   minor(rowNum, colNum) {
       if (!isSquare) Fiber.abort("Matrix must be square.")
       if (!validRowNum_(rowNum)) Fiber.abort("Invalid row number.")
       if (!validColNum_(colNum)) Fiber.abort("Invalid column number.")
       return minor_(rowNum, colNum)
   }

   // Private version of the above method which returns the minor without
   // validity checks.
   minor_(x, y) {
       var len = _nr - 1
       var result = List.filled(len, null)
       for (i in 0...len) {
           result[i] = List.filled(len, null)
           for (j in 0...len) {
               if (i < x && j < y) {
                   result[i][j] = _a[i][j]
               } else if (i >= x && j < y) {
                   result[i][j] = _a[i+1][j]
               } else if (i < x && j >= y) {
                   result[i][j] = _a[i][j+1]
               } else {
                   result[i][j] = _a[i+1][j+1]
               }
           }
       }
       return CMatrix.new_(result)
   }

   // Returns the complex matrix of cofactors of the current instance.
   cofactors {
       if (!isSquare) Fiber.abort("Matrix must be square.")
       var cf = List.filled(_nr, null)
       for (i in 0..._nr) {
           cf[i] = List.filled(_nc, null)
           for (j in 0..._nc)  cf[i][j] = minor_(i, j).det * (Complex.minusOne.pow(i + j))
       }
       return CMatrix.new_(cf)
   }

   // Returns the adjugate of the current instance.
   adjugate { cofactors.transpose }

   // Returns the inverse of this instance if it's square and if it exists
   // using the Gauss-Jordan method.
   inverse {
       if (!isSquare) Fiber.abort("Matrix must be square.")
       if (det == Complex.zero) Fiber.abort("No inverse as determinant is zero.")
       var aug = CMatrix.new_(_nr, 2 *_nr, Complex.zero)
       for (i in 0..._nr) {
           for (j in 0..._nr) aug.set_(i, j, _a[i][j])
           aug.set_(i, i + _nr, Complex.one)
       }
       aug.toReducedRowEchelonForm
       var inv = CMatrix.new_(_nr, _nr, Complex.zero)
       for (i in 0..._nr) {
           for (j in _nr...2 *_nr) inv.set_(i, j - _nr, aug.get_(i, j))
       }
       return inv
   }

   // Converts the current instance in place to reduced row echelon form.
   toReducedRowEchelonForm {
       var lead = 0
       for (r in 0..._nr) {
           if (_nc <= lead) return
           var i = r
           while (_a[i][lead] == Complex.zero) {
               i = i + 1
               if (_nr == i) {
                   i = r
                   lead = lead + 1
                   if (_nc == lead) return
               }
           }
           swapRows_(i, r)
           if (_a[r][lead] != Complex.zero) {
               var div = _a[r][lead]
               for (j in 0..._nc) _a[r][j] = _a[r][j] / div
           }
           for (k in 0..._nr) {
               if (k != r) {
                   var mult = _a[k][lead]
                   for (j in 0..._nc) _a[k][j] = _a[k][j] - _a[r][j] * mult
               }
           }
           lead = lead + 1
       }
   }

   // Create a new submatrix from rowNum1 to rowNum2 inclusive and from
   // colNum1 to colNum2 inclusive of the current instance.
   subMatrix(rowNum1, colNum1, rowNum2, colNum2) {
       if (!validRowNum_(rowNum1)) Fiber.abort("Invalid first row number.")
       if (!validColNum_(colNum1)) Fiber.abort("Invalid first column number.")
       if (!validRowNum_(rowNum2)) Fiber.abort("Invalid second row number.")
       if (!validColNum_(colNum2)) Fiber.abort("Invalid second column number.")
       if (rowNum1 > rowNum2) Fiber.abort("First row number cannot be greater than second.")
       if (colNum1 > colNum2) Fiber.abort("First column number cannot be greater than second.")
       return subMatrix_(rowNum1, colNum1, rowNum2, colNum2)
   }

   // Private version of the above method which returns the submatrix without
   // validity checks.
   subMatrix_(rowNum1, colNum1, rowNum2, colNum2) {
       var t = CMatrix.new_(rowNum2 - rowNum1 + 1, colNum2 - colNum1 + 1, Complex.zero)
       for (i in rowNum1..rowNum2) {
           for (j in colNum1..colNum2) {
               t.set_(i - rowNum1, j - colNum1, _a[i][j])
           }
       }
       return t
   }

   // Returns the trace of the current instance if it's square.
   trace {
       if (!isSquare) Fiber.abort("Cannot calculate the trace of a non-square matrix.")
       var sum = Complex.zero
       for (i in 0..._nr) sum = sum + _a[i][i]
       return sum
   }

   // Returns the determinant of the current instance if it's square using
   // Laplace expansion.
   det {
       if (!isSquare) Fiber.abort("Cannot calculate the determinant of a non-square matrix.")
       if (_nr == 1) return _a[0][0]
       if (_nr == 2) return _a[1][1] * _a[0][0] - _a[0][1] * _a[1][0]
       var sign = Complex.one
       var sum = Complex.zero
       for (i in 0..._nr) {
           var m = minor_(0, i)
           sum = sum + sign * _a[0][i] * m.det
           sign = -sign
       }
       return sum
   }

   // Returns the permanent of the current instance if it's square using
   // Laplace expansion.
   perm {
       if (!isSquare) Fiber.abort("Cannot calculate the permanent of a non-square matrix.")
       if (_nr == 1) return _a[0][0]
       var sum = Complex.zero
       for (i in 0..._nr) {
           var m = minor_(0, i)
           sum = sum + _a[0][i] * m.perm
       }
       return sum
   }

   // Returns the sum of all elements of the current instance.
   sum {
       var sum = Complex.zero
       for (i in 0..._nr) {
           for (j in 0..._nc) sum = sum + _a[i][j]
       }
       return sum
   }

   // Returns the norm of all elements of the current instance.
   norm {
       var sum = Complex.zero
       for (i in 0..._nr) {
           for (j in 0..._nc) sum = sum + _a[i][j] * _a[i][j]
       }
       return sum.sqrt
   }

   // Returns the product of all elements of the current instance.
   prod {
       var prd = Complex.one
       for (i in 0..._nr) {
           for (j in 0..._nc) {
               if (_a[i][j] == Complex.zero) return Complex.zero
               prd = prd * _a[i][j]
           }
       }
       return prd
   }

   // Prints the current instance's elements as a 2D list with each row on a new line.
   print() { System.print(_a.join("\n")) }

   // Returns the current instance's elements as a string.
   toString { _a.toString }

}

/* CMatrices contains various routines applicable to lists of CMatrix objects. */ class CMatrices {

   static sum(a)  { a.reduce { |acc, x| acc + x } }
   static prod(a) { a[1..-1].reduce(a[0]) { |acc, x| acc * x } }

}

// Type aliases for classes in case of any name clashes with other modules. var Complex_Complex = Complex var Complex_Complexes = Complexes var Complex_CMatrix = CMatrix var Complex_CMatrices = CMatrices var Complex_Cloneable = Cloneable // in case imported indirectly</lang>