# 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

```/* 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 three        { Complex.new_( 3,  0) }
static four         { Complex.new_( 4,  0) }
static five         { Complex.new_( 5,  0) }
static six          { Complex.new_( 6,  0) }
static seven        { Complex.new_( 7,  0) }
static eight        { Complex.new_( 8,  0) }
static nine         { Complex.new_( 9,  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 imagThree    { Complex.new_( 0,  3) }
static imagFour     { Complex.new_( 0,  4) }
static imagFive     { Complex.new_( 0,  5) }
static imagSix      { Complex.new_( 0,  6) }
static imagSeven    { Complex.new_( 0,  7) }
static imagEight    { Complex.new_( 0,  8) }
static imagNine     { Complex.new_( 0,  9) }
static imagTen      { Complex.new_( 0, 10) }
static i            { Complex.new_( 0,  1) } // same as imagOne

static pi           { Complex.new_(Num.pi, 0) }
static tau          { Complex.new_(Num.tau, 0) }
static e            { Complex.new_(2.71828182845904523536, 0) }
static phi          { Complex.new_(1.6180339887498948482,  0) } // golden ratio
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'
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 = []
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.

// 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 {
for (r in 0..._nr) {
var i = r
i = i + 1
if (_nr == i) {
i = r
}
}
swapRows_(i, r)
for (j in 0..._nc) _a[r][j] = _a[r][j] / div
}
for (k in 0..._nr) {
if (k != r) {
for (j in 0..._nc) _a[k][j] = _a[k][j] - _a[r][j] * mult
}
}
}
}

// 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 } }
}
```