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‎utils/Function: Complex Numbers.pinescript

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// This source code is subject to the terms of the Mozilla Public License 2.0 at https://mozilla.org/MPL/2.0/
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// © RicardoSantos
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//@version=4
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study(title='Function: Complex Numbers')
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//| Complex numbers prototype: {
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//|----------------------------------------------------------------------------||
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//|----------------------------------------------------------------------------||
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//|----------------------------------------------------------------------------||
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//| @description:
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//| A set of utility functions to handle complex numbers.
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//| @references:
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//| https://en.wikipedia.org/wiki/Complex_number
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//| js: https://rosettacode.org/wiki/Fast_Fourier_transform
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//| https://github.com/trekhleb/javascript-algorithms/blob/477f30b0bdac6024874d2976de1eca7afe0176dd/src/algorithms/math/complex-number/ComplexNumber.js#L3
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//| https://github.com/infusion/Complex.js/
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//| http://jmvidal.cse.sc.edu/talks/javascript/complexnumbersexample.html
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//|----------------------------------------------------------------------------||
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//|----------------------------------------------------------------------------||
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//|----------------------------------------------------------------------------||
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f_complex_new(_real, _imaginary)=>
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//| @description:
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//| Creates a prototype array to handle complex numbers.
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//| @parameters:
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//| _real: real value of the complex number.
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//| _imaginary: imaginary number of the complex number. optional(use float(na) to not define value)
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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array.set(id=_complex, index=0, value=_real)
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array.set(id=_complex, index=1, value=na(_imaginary)?0.0:_imaginary)
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary].
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_complex
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f_complex_real_get(_a)=>
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//| @description:
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//| Get the real part of complex number _a.
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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_real = array.get(id=_a, index=0)
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//| @return:
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//| _real: Real part of the complex number _a.
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_real
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f_complex_imaginary_get(_a)=>
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//| @description:
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//| Get the imaginary part of complex number _a.
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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_imaginary = array.get(id=_a, index=1)
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//| @return:
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//| _imaginary: Imaginary part of the complex number _a.
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_imaginary
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f_complex_add(_a, _b)=>
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//| @description:
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//| Adds complex number _b to _a, in the form:
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//| [_a.real + _b.real, _a.imaginary + _b.imaginary].
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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//| _b: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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array.set(id=_complex, index=0, value=f_complex_real_get(_a) + f_complex_real_get(_b))
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array.set(id=_complex, index=1, value=f_complex_imaginary_get(_a) + f_complex_imaginary_get(_b))
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_subtract(_a, _b)=>
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//| @description:
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//| Subtract complex number _b from _a, in the form:
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//| [_a.real - _b.real, _a.imaginary - _b.imaginary].
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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//| _b: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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array.set(id=_complex, index=0, value=f_complex_real_get(_a) - f_complex_real_get(_b))
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array.set(id=_complex, index=1, value=f_complex_imaginary_get(_a) - f_complex_imaginary_get(_b))
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_multiply(_a, _b)=>
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//| @description:
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//| Multiply complex number _a with _b, in the form:
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//| [(_a.real * _b.real) - (_a.imaginary * _b.imaginary), (_a.real * _b.imaginary) + (_a.imaginary * _b.real)]
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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//| _b: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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float _b_real = f_complex_real_get(_b)
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float _b_imaginary = f_complex_imaginary_get(_b)
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float _r = _a_real * _b_real - _a_imaginary * _b_imaginary
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float _i = _a_real * _b_imaginary + _a_imaginary * _b_real
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array.set(id=_complex, index=0, value=_r)
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array.set(id=_complex, index=1, value=_i)
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_negative(_a)=>
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//| @description:
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//| Negative of complex number _a, in the form: [-_a.real, -_a.imaginary]
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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array.set(id=_complex, index=0, value=-_a_real)
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array.set(id=_complex, index=1, value=-_a_imaginary)
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_cexp(_a)=>
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//| @description:
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//| Exponential of complex number _a, in the form:
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//| reference: https://en.cppreference.com/w/c/numeric/complex/cexp
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//| http://cboard.cprogramming.com/c-programming/89116-how-implement-complex-exponential-functions-c.html#post637921
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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var _er = exp(_a_real)
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array.set(id=_complex, index=0, value=_er * cos(_a_imaginary))
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array.set(id=_complex, index=1, value=_er * sin(_a_imaginary))
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_tostring(_a)=>
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//| @description:
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//| Converts complex number _a to a string format, in the form: "a+bi"
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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string _str = ''
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if na(_a_real)
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_str := tostring(_a_imaginary) + 'i'
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else if _a_imaginary < 0
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_str := tostring(_a_real) + '' + tostring(_a_imaginary) + 'i'
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else
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_str := tostring(_a_real) + '+' + tostring(_a_imaginary) + 'i'
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//| @return:
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//| _str: a string int "a+bi" format
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_str
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f_complex_conjugate(_a)=>
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//| @description:
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//| Computes the conjugate of complex number _a by reversing the sign of the imaginary part.
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//| reference: https://en.cppreference.com/w/c/numeric/complex/conj
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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float[] _complex = f_complex_new(_a_real, -1 * _a_imaginary)
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_divide(_a, _b)=>
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//| @description:
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//| Divide complex number _a with _b, in the form:
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//| [(_a.real * _b.real) - (_a.imaginary * _b.imaginary), (_a.real * _b.imaginary) + (_a.imaginary * _b.real)]
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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//| _b: pseudo complex number in the form of a array [real, imaginary].
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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float _b_real = f_complex_real_get(_b)
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float _b_imaginary = f_complex_imaginary_get(_b)
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// Make sure we're dealing with complex number.
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float[] _complexDivider = f_complex_new(_b_real, _b_imaginary)
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// Multiply dividend by divider's conjugate.
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float[] _finalDivident = f_complex_multiply(_a, f_complex_conjugate(_complexDivider))
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// Calculating final divider using formula (a + bi)(a − bi) = a^2 + b^2
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float finalDivider = pow(f_complex_real_get(_complexDivider), 2) + pow(f_complex_imaginary_get(_complexDivider), 2)
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float _finaldivident_real = f_complex_real_get(_finalDivident)
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float _finaldivident_imaginary = f_complex_imaginary_get(_finalDivident)
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array.set(id=_complex, index=0, value=_finaldivident_real / finalDivider)
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array.set(id=_complex, index=1, value=_finaldivident_imaginary / finalDivider)
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_get_radius(_a)=>
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//| @description:
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//| Radius of _a, in the form: [sqrt(pow(_a))]
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//| Return the magnitude of a complex number. This is defined
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//| as its distance from the origin (0,0) of the complex plane.
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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float _radius= sqrt(pow(f_complex_real_get(_a), 2) + pow(f_complex_imaginary_get(_a), 2))
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//| @return:
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//| _radius: float value with radius.
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_radius
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f_complex_get_phase(_a, _inRadians)=>
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//| @description:
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//| The phase value of complex number _a.
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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//| _inRadians: boolean value for the type of angle value, (true: radians, false: degrees)
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float _pi = 3.14159265359
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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float _phase = atan(abs(_a_imaginary) / abs(_a_real))
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if not _inRadians
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_phase := _phase * (180 / _pi) // radian to degrees
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else
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if _a_real < 0 and _a_imaginary > 0
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_phase := _pi - nz(_phase)
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else if _a_real < 0 and _a_imaginary < 0
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_phase := -(_pi - nz(_phase))
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else if _a_real > 0 and _a_imaginary < 0
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_phase := -nz(_phase)
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else if _a_real == 0 and _a_imaginary > 0
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_phase := _pi / 2.0
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else if _a_real == 0 and _a_imaginary < 0
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_phase := -_pi / 2.0
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else if _a_real < 0 and _a_imaginary == 0
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_phase := _pi
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else if _a_real > 0 and _a_imaginary == 0
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_phase := 0.0
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else if _a_real == 0 and _a_imaginary == 0
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// More correctly would be to set 'indeterminate'.
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// But just for simplicity reasons let's set zero.
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_phase := 0.0
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else
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na
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//| @return:
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//| _phase: float value with phase.
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_phase
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f_complex_get_polar_form(_a, _inRadians)=>
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//| @description:
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//| The polar form value of complex number _a.
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//| @parameters:
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//| _a: pseudo complex number in the form of a array [real, imaginary].
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//| _inRadians: boolean value for the type of angle value, (true: radians, false: degrees)
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float[] _complex = f_complex_new(f_complex_get_radius(_a), f_complex_get_phase(_a, _inRadians))
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//| @return:
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//| _complex: **pseudo complex number in the form of a array [real, imaginary]
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//| ** returns a array [radius, phase]
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_complex
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f_complex_array_new(_size, _initial_complex)=>
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//| @description:
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//| Prototype to initialize a array of complex numbers.
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//| @parameters:
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//| _size: size of the array.
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//| _initial_complex: Complex number to be used as default value, in the form of array [real, imaginary].
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float _init_real = array.get(id=_initial_complex, index=0)
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float _init_imaginary = array.get(id=_initial_complex, index=1)
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float[] _complex_array = array.new_float(0)
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for _i = 0 to _size-1
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array.push(id=_complex_array, value=_init_real)
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array.push(id=_complex_array, value=_init_imaginary)
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//| @return:
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//| _complex_array: pseudo complex Array in the form of a array [0:real, 1:imaginary, 2:real, 3:imaginary,...]
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_complex_array
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f_complex_array_get(_id, _index)=>
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//| @description:
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//| Get the complex number in a array, in the form of a array [real, imaginary]
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//| @parameters:
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//| _id: ID of the array.
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//| _index: Index of the complex number.
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float[] _complex = array.new_float(size=2, initial_value=0.0)
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int _size = round(array.size(id=_id) / 2)
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int _index2 = _index * 2
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if _index > -1 and _index < _size
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array.set(id=_complex, index=0, value=array.get(id=_id, index=_index2))
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array.set(id=_complex, index=1, value=array.get(id=_id, index=_index2 + 1))
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//| @return:
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//| _complex: pseudo complex number in the form of a array [real, imaginary]
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_complex
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f_complex_array_set(_id, _index, _a)=>
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//| @description:
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//| Sets the values complex number in a array.
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//| @parameters:
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//| _id: ID of the array.
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//| _index: Index of the complex number.
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//| _a: Complex number, in the form: [real, imaginary].
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float _a_real = f_complex_real_get(_a)
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float _a_imaginary = f_complex_imaginary_get(_a)
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int _size = array.size(id=_id)-1
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int _index2 = _index * 2
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if _index > -1 and _index2 < _size
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array.set(id=_id, index=_index2, value=_a_real)
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array.set(id=_id, index=_index2 + 1, value=_a_imaginary)
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//| @return: Void, updates array _id.
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f_complex_array_tostring(_id)=>
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//| @description:
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//| Reads a array of complex numbers into a string, of the form: "[ [a+bi], ... ]""
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//| @parameters:
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//| _id: ID of the array.
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int _size = round(array.size(id=_id) / 2)
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string _str = '[ '
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if _size > 0
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for _i = 0 to _size-1
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_str := _str + '[ ' + f_complex_tostring(f_complex_array_get(_id, _i)) + ' ]'
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if _i == _size
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_str := _str + ' ]'
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else
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_str := _str + ', '
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//| @return:
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//| _str: Translated complex array into string.
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_str
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//|----------------------------------------------------------------------------||
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//|--------------------------------}-------------------------------------------||
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//|----------------------------------------------------------------------------||
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//|----------------------------------------------------------------------------||
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//|----------------------------------------------------------------------------||
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if barstate.ishistory[1] and barstate.isrealtime
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float[] a = f_complex_new(0.10, -0.25)
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float[] b = f_complex_new(0.66, float(na))
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float[] c = f_complex_add(a, b)
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float[] d = f_complex_subtract(c, b)
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float[] e = f_complex_multiply(a, d)
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float[] f = f_complex_cexp(e)
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float[] g = f_complex_conjugate(f)
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float[] h = f_complex_divide(f, g)
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float i = f_complex_get_radius(h)
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float j = f_complex_get_phase(h, true)
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float[] k = f_complex_get_polar_form(h, true)
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float[] L = f_complex_array_new(4, f_complex_new(0, 0))
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f_complex_array_set(L, 0, a)
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f_complex_array_set(L, 1, b)
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f_complex_array_set(L, 2, c)
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f_complex_array_set(L, 3, d)
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float[] m = f_complex_array_get(L, 2)
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float[] n = f_complex_negative(m)
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_text = ''
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_text := _text + 'a: ' + f_complex_tostring(a) + '\n'
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_text := _text + 'b: ' + f_complex_tostring(b) + '\n'
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_text := _text + 'c: ' + f_complex_tostring(c) + '\n'
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_text := _text + 'd: ' + f_complex_tostring(d) + '\n'
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_text := _text + 'e: ' + f_complex_tostring(e) + '\n'
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_text := _text + 'f: ' + f_complex_tostring(f) + '\n'
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_text := _text + 'g: ' + f_complex_tostring(g) + '\n'
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_text := _text + 'h: ' + f_complex_tostring(h) + '\n'
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_text := _text + 'i: ' + tostring(i) + '\n'
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_text := _text + 'j: ' + tostring(j) + '\n'
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_text := _text + 'k: ' + tostring(k) + '\n'
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_text := _text + 'L: ' + f_complex_array_tostring(L) + '\n'
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_text := _text + 'm: ' + f_complex_tostring(m) + '\n'
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_text := _text + 'n: ' + f_complex_tostring(n) + '\n'
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label.new(x=bar_index, y=0.0, text=_text, textalign=text.align_left)
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