621 lines
21 KiB
MQL5
621 lines
21 KiB
MQL5
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//+------------------------------------------------------------------+
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//| cmath.mqh |
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//| Copyright 2025, MetaQuotes Ltd. |
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//| https://www.mql5.com |
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//+------------------------------------------------------------------+
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#property copyright "Copyright 2025, MetaQuotes Ltd."
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#property link "https://www.mql5.com"
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//+------------------------------------------------------------------+
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//| utilities for working with complex numbers |
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//+------------------------------------------------------------------+
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//+------------------------------------------------------------------+
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//| get real part from container of complex numbers |
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//+------------------------------------------------------------------+
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void real(complex& data[],double& out[])
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{
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// Resize the output array to match the source array size
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ArrayResize(out,data.Size());
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// Extract the real component element-by-element
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for(ulong i = 0; i<out.Size(); i++)
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out[i] = data[i].real;
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}
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//+------------------------------------------------------------------+
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//| get imaginary part from container of complex numbers |
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//+------------------------------------------------------------------+
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void imaginary(complex& data[],double& out[])
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{
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// Resize the output array to match the source array size
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ArrayResize(out,data.Size());
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// Extract the imaginary component element-by-element
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for(ulong i = 0; i<out.Size(); i++)
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out[i] = data[i].imag;
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}
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//+------------------------------------------------------------------+
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//| get real part from container of complex numbers |
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//+------------------------------------------------------------------+
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vector real(vectorc& data)
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{
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// Initialize an output vector of the same size with zeros
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vector out=vector::Zeros(data.Size());
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for(ulong i = 0; i<out.Size(); i++)
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out[i] = data[i].real;
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return out;
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}
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//+------------------------------------------------------------------+
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//| get imaginary part from container of complex numbers |
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//+------------------------------------------------------------------+
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vector imaginary(vectorc& data)
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{
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// Initialize an output vector of the same size with zeros
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vector out=vector::Zeros(data.Size());
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for(ulong i = 0; i<out.Size(); i++)
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out[i] = data[i].imag;
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return out;
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}
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//+------------------------------------------------------------------+
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//| get real part from container of complex numbers |
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//+------------------------------------------------------------------+
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matrix real(matrixc& data)
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{
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// Initialize an output matrix matching dimensions with zeros
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matrix out=matrix::Zeros(data.Rows(),data.Cols());
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// Iterate through rows and columns to copy real values
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for(ulong i = 0; i<out.Rows(); i++)
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for(ulong j = 0; j<out.Cols(); j++)
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out[i,j] = data[i,j].real;
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return out;
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}
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//+------------------------------------------------------------------+
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//| get imaginary part from container of complex numbers |
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//+------------------------------------------------------------------+
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matrix imaginary(matrixc& data)
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{
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// Initialize an output matrix matching dimensions with zeros
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matrix out=matrix::Zeros(data.Rows(),data.Cols());
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// Iterate through rows and columns to copy imaginary values
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for(ulong i = 0; i<out.Rows(); i++)
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for(ulong j = 0; j<out.Cols(); j++)
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out[i,j] = data[i,j].imag;
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return out;
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}
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//+------------------------------------------------------------------+
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//| create complex containers from real numbers |
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//+------------------------------------------------------------------+
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vectorc as_complex(vector& re,vector& im)
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{
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// Size output based on the largest input vector to prevent truncation
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vectorc out = vectorc::Zeros(MathMax(re.Size(),im.Size()));
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for(ulong i = 0; i<out.Size(); i++)
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{
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// Fallback to 0.0 if index exceeds the size of an uneven input vector
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out[i].real = i<re.Size()?re[i]:0.0;
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out[i].imag = i<im.Size()?im[i]:0.0;
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}
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return out;
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}
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//+------------------------------------------------------------------+
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//| create complex containers from real numbers |
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//+------------------------------------------------------------------+
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matrixc as_complex(matrix& re,matrix& im)
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{
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// Size output based on the maximum bounds of both matrices
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matrixc out = matrixc::Zeros((ulong)MathMax(re.Rows(),im.Rows()),(ulong)MathMax(re.Cols(),im.Cols()));
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for(ulong i = 0; i<out.Rows(); i++)
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{
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for(ulong j = 0; j<out.Cols(); j++)
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{
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// Safely extract real and imaginary values within bound checks
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out[i,j].real = (i<re.Rows() && j<re.Cols())?re[i,j]:0.0;
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out[i,j].imag = (i<im.Rows() && j<im.Cols())?im[i,j]:0.0;
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}
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}
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return out;
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}
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//+------------------------------------------------------------------+
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//| create complex containers from real numbers |
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//+------------------------------------------------------------------+
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vectorc as_complex(vector& re)
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{
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vectorc out = vectorc::Zeros(re.Size());
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for(ulong i = 0; i<out.Size(); i++)
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{
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out[i].real = re[i];
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out[i].imag = 0.0; // Imaginary part defaults to zero
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}
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return out;
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}
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//+------------------------------------------------------------------+
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//| create complex containers from real numbers |
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//+------------------------------------------------------------------+
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matrixc as_complex(matrix& re)
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{
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matrixc out = matrixc::Zeros(re.Rows(),re.Cols());
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for(ulong i = 0; i<out.Rows(); i++)
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{
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for(ulong j = 0; j<out.Cols(); j++)
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{
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out[i,j].real = re[i,j];
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out[i,j].imag = 0.0; // Imaginary part defaults to zero
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}
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}
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return out;
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}
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//+------------------------------------------------------------------+
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//| struct for complex vectors |
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//+------------------------------------------------------------------+
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struct CComplexVector
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{
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private:
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// Unpack vectorc structured data into separate real and imaginary vectors
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void to_real(vectorc& data)
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{
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im = re = vector::Zeros(data.Size());
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for(ulong i = 0; i<data.Size(); i++)
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{
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re[i] = data[i].real;
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im[i] = data[i].imag;
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}
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}
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// Convert purely real vector input by initializing the imaginary vector to zero
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void to_complex(vector& in)
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{
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im = vector::Zeros(in.Size());
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re = in;
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}
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// Unpack array of complex structures into internal real/imaginary vectors
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void to_real(complex & data[])
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{
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im = re = vector::Zeros(data.Size());
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for(uint i = 0; i<data.Size(); i++)
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{
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re[i] = data[i].real;
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im[i] = data[i].imag;
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}
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}
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// Convert double array input by initializing internal vectors and assigning values
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void to_complex(double& in[])
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{
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re = im = vector::Zeros(in.Size());
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re.Assign(in);
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}
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// Reset vector allocations to zero size
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void reset(void)
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{
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re = vector::Zeros(0);
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im = vector::Zeros(0);
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}
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public:
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vector re; // Holds real parts
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vector im; // Holds imaginary parts
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// Default Constructor
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CComplexVector(void)
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{
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re = vector::Zeros(0);
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im = vector::Zeros(0);
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}
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// Overloaded Constructors
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CComplexVector(vectorc& complx) { to_real(complx); }
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CComplexVector(vector& realv) { to_complex(realv); }
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CComplexVector(CComplexVector& other)
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{
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re = other.re;
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im = other.im;
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}
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// Overloaded index operator to retrieve elements as a singular complex structure
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complex operator[](ulong index)
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{
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complex out=0;
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if(index<re.Size())
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{
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out.real = re[index];
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out.imag = im[index];
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return out;
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}
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else
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{
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Print(__FUNCTION__, " invalid index ", index, " : container size ", re.Size());
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return out;
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}
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}
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// Assignment Operators
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void operator=(vectorc& complex_vector) { to_real(complex_vector); }
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void operator=(vector& real_vector) { to_complex(real_vector); }
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void operator=(complex& complex_array[]) { to_real(complex_array); }
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void operator=(double& real_array[]) { to_complex(real_array); }
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// Convert back to native vectorc format
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vectorc to_vector(void) { return as_complex(re,im); }
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};
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//+------------------------------------------------------------------+
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//| struct for complex matrices |
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//+------------------------------------------------------------------+
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struct CComplexMatrix
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{
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private:
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// Unpack matrixc structure into separate internal real and imaginary matrices
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void to_real(matrixc& data)
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{
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im = re = matrix::Zeros(data.Rows(),data.Cols());
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for(ulong i = 0; i<data.Rows(); i++)
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{
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for(ulong j = 0; j<data.Cols(); j++)
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{
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re[i,j] = data[i][j].real;
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im[i,j] = data[i][j].imag;
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}
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}
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}
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// Convert purely real matrix input by zeroing out the imaginary matrix
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void to_complex(matrix& in)
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{
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im = matrix::Zeros(in.Rows(),in.Cols());
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re = in;
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}
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// Clear memory footprint of internal matrices
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void reset(void)
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{
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re = matrix::Zeros(0,0);
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im = matrix::Zeros(0,0);
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}
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public:
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matrix re; // Holds real matrix space
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matrix im; // Holds imaginary matrix space
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// Default Constructor
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CComplexMatrix(void)
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{
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re = matrix::Zeros(0,0);
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im = matrix::Zeros(0,0);
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}
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// Overloaded Constructors
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CComplexMatrix(matrixc& complx) { to_real(complx); }
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CComplexMatrix(matrix& realm) { to_complex(realm); }
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CComplexMatrix(CComplexMatrix& other)
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{
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re = other.re;
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im = other.im;
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}
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// Safe element retrieval at explicit row/column coordinates
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complex At(ulong row_index,ulong column_index)
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{
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complex out = 0;
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if(row_index<re.Rows() && column_index<re.Cols())
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{
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out.real = re[row_index,column_index];
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out.imag = im[row_index,column_index];
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return out;
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}
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else
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{
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Print(__FUNCTION__, " invalid row or column index ", " container rows ", re.Rows(), " container columns ", re.Cols());
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return complex(0.0);
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}
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}
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// Assignment Operators
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void operator=(matrixc& complex_matrix) { to_real(complex_matrix); }
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void operator=(matrix& real_matrix) { to_complex(real_matrix); }
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// Convert back to native matrixc format
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matrixc to_matrix(void) { return as_complex(re,im); }
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};
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//+------------------------------------------------------------------+
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//| power of complex number |
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//+------------------------------------------------------------------+
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complex c_pow(complex base, complex exponent)
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{
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// Calculated via identity: base^exponent = exp(exponent * ln(base))
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return c_exp(exponent*c_log(base));
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}
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//+------------------------------------------------------------------+
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//| simple power of complex number |
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//+------------------------------------------------------------------+
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vectorc c_pow(vectorc &vect, complex exponent)
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{
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vectorc out(vect.Size());
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for(ulong i = 0; i<vect.Size(); i++)
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out[i] = c_pow(vect[i],exponent);
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return out;
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}
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//+------------------------------------------------------------------+
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//| simple power of complex number |
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//+------------------------------------------------------------------+
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matrixc c_pow(matrixc &matrx, complex exponent)
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{
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matrixc out(matrx.Rows(),matrx.Cols());
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// Process column by column using vector-based power calculations
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for(ulong i = 0; i<out.Cols(); i++)
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out.Col(c_pow(matrx.Col(i),exponent),i);
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return out;
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}
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//+------------------------------------------------------------------+
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//| power of complex number |
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//+------------------------------------------------------------------+
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complex c_pow(complex value, double alpha)
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{
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complex out = value;
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double r = c_abs(value); // Magnitude
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double theta = atan2(value.imag,value.real); // Phase Angle
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double n_r = pow(r,alpha); // New Magnitude
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double n_theta = alpha*theta; // New Phase Angle
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// Convert back from polar form to rectangular form
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out.real = n_r*cos(n_theta);
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out.imag = n_r*sin(n_theta);
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return out;
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}
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//+------------------------------------------------------------------+
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//| simple power of complex number |
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//+------------------------------------------------------------------+
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vectorc c_pow(vectorc &vect, double alpha)
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{
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vectorc out(vect.Size());
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for(ulong i = 0; i<vect.Size(); i++)
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out[i] = c_pow(vect[i],alpha);
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||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| simple power of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
matrixc c_pow(matrixc &matrx, double alpha)
|
||
|
|
{
|
||
|
|
matrixc out(matrx.Rows(),matrx.Cols());
|
||
|
|
// Process matrix columns individually
|
||
|
|
for(ulong i = 0; i<out.Cols(); i++)
|
||
|
|
out.Col(c_pow(matrx.Col(i),alpha),i);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Complex Signum Function |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
complex signum(complex z)
|
||
|
|
{
|
||
|
|
double magnitude = sqrt((z.real*z.real) + (z.imag*z.imag));
|
||
|
|
if(!magnitude)
|
||
|
|
return 0+0i; // Handle division by zero for origin coordinates
|
||
|
|
|
||
|
|
complex magz;
|
||
|
|
magz.real = magnitude;
|
||
|
|
magz.imag = 0.0;
|
||
|
|
|
||
|
|
// Returns unit vector directed in the same phase angle (z / |z|)
|
||
|
|
return z/magz;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| absolute value of a complex number (magnitude) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
double c_abs(complex z)
|
||
|
|
{
|
||
|
|
return sqrt((z.real*z.real) + (z.imag*z.imag));
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| absolute value of a complex number (magnitude) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
vector c_abs(vectorc &z)
|
||
|
|
{
|
||
|
|
CComplexVector v(z);
|
||
|
|
// Vectorized calculation of Euclidean length
|
||
|
|
return sqrt((v.re*v.re) + (v.im*v.im));
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| absolute value of a complex number (magnitude) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
matrix abs(matrixc &z)
|
||
|
|
{
|
||
|
|
CComplexMatrix v(z);
|
||
|
|
// Matrix-wide vectorized calculation of absolute values
|
||
|
|
return sqrt((v.re*v.re) + (v.im*v.im));
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| sqrt of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
complex c_sqrt(complex z)
|
||
|
|
{
|
||
|
|
double mag_z = c_abs(z);
|
||
|
|
mag_z=sqrt(mag_z); // Square root of the magnitude
|
||
|
|
double theta = atan2(z.imag,z.real); // Phase Angle
|
||
|
|
complex out;
|
||
|
|
// Half-angle principles for complex square root calculation
|
||
|
|
out.real = mag_z*cos(theta/2.0);
|
||
|
|
out.imag = mag_z*sin(theta/2.0);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| sqrt of complex vector |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
vectorc c_sqrt(vectorc &z)
|
||
|
|
{
|
||
|
|
vectorc out = z;
|
||
|
|
for(ulong i = 0; i<out.Size(); i++)
|
||
|
|
out[i] = c_sqrt(z[i]);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| sqrt of complex vector |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
matrixc c_sqrt(matrixc &z)
|
||
|
|
{
|
||
|
|
matrixc out = z;
|
||
|
|
for(ulong i = 0; i<out.Cols(); i++)
|
||
|
|
out.Col(c_sqrt(z.Col(i)),i);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| log of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
complex c_log(complex z)
|
||
|
|
{
|
||
|
|
complex out;
|
||
|
|
// Natural logarithm of complex number: ln(z) = ln(|z|) + i*theta
|
||
|
|
out.real = c_abs(z);
|
||
|
|
out.imag = atan2(z.imag,z.real);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| log of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
vectorc c_log(vectorc &z)
|
||
|
|
{
|
||
|
|
vectorc out = z;
|
||
|
|
for(ulong i = 0; i<out.Size(); i++)
|
||
|
|
out[i] = c_log(z[i]);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| log of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
matrixc c_log(matrixc &z)
|
||
|
|
{
|
||
|
|
matrixc out = z;
|
||
|
|
for(ulong i = 0; i<out.Cols(); i++)
|
||
|
|
out.Col(c_log(z.Col(i)),i);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| exp function of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
complex c_exp(complex z)
|
||
|
|
{
|
||
|
|
complex out;
|
||
|
|
// Euler's Formula conversion: exp(x + iy) = exp(x) * (cos(y) + i*sin(y))
|
||
|
|
out.real = exp(z.real)*cos(z.imag);
|
||
|
|
out.imag = exp(z.real)*sin(z.imag);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| exp function of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
vectorc c_exp(vectorc &z)
|
||
|
|
{
|
||
|
|
vectorc out = z;
|
||
|
|
for(ulong i = 0; i<out.Size(); i++)
|
||
|
|
out[i] = c_exp(z[i]);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| exp function of complex number |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
matrixc c_exp(matrixc &z)
|
||
|
|
{
|
||
|
|
matrixc out = z;
|
||
|
|
for(ulong i = 0; i<out.Cols(); i++)
|
||
|
|
out.Col(c_exp(z.Col(i)),i);
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+-----------------------------------------------------------------------+
|
||
|
|
//| Generate a complex Vandermonde matrix |
|
||
|
|
//| in : vectorc |
|
||
|
|
//| num_columns : ulong, default: size of in input vector. |
|
||
|
|
//| ascending : bool, (default: false) Order of the powers of the columns.|
|
||
|
|
//| If true, the powers increase |
|
||
|
|
//| from left to right, if false (the default) they are reversed |
|
||
|
|
//+-----------------------------------------------------------------------+
|
||
|
|
matrixc vanderc(vectorc &in, ulong num_columns = 0,bool ascending = false)
|
||
|
|
{
|
||
|
|
ulong n = num_columns?num_columns:in.Size();
|
||
|
|
|
||
|
|
matrixc out(in.Size(),n);
|
||
|
|
|
||
|
|
// Populate columns by geometric power progression rules specified by 'ascending'
|
||
|
|
for(ulong i = 0; i<n; i++)
|
||
|
|
out.Col(c_pow(in,ascending?double(i):double(n-1-i)),i);
|
||
|
|
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//|process vector representing complex numbers in compact form |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
vectorc compact_to_complex(vector& v,vectorc& bp)
|
||
|
|
{
|
||
|
|
vectorc v_out = vectorc::Zeros(v.Size());
|
||
|
|
|
||
|
|
for(ulong i = 0; i<v.Size() && i<bp.Size();)
|
||
|
|
{
|
||
|
|
// If the blueprint tracking parameter has an imaginary element, parse conjugate pairs
|
||
|
|
if(bp[i].imag)
|
||
|
|
{
|
||
|
|
v_out[i].real = v[i];
|
||
|
|
if(i+1 < v.Size())
|
||
|
|
{
|
||
|
|
// Maps the pair out as real + i*imag and real - i*imag sequential indexes
|
||
|
|
v_out[i].imag = v[i+1];
|
||
|
|
v_out[i+1].real = v_out[i].real;
|
||
|
|
v_out[i+1].imag = -1.0*v_out[i].imag; // Conjugate inversion
|
||
|
|
i+=2; // Step over both elements processed
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// Purely real number fallback step
|
||
|
|
v_out[i].real = v[i];
|
||
|
|
i+=1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
return v_out;
|
||
|
|
}
|
||
|
|
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| process matrix representing complex numbers in compact form |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
matrixc compact_to_complex(matrix& m, vectorc& bp)
|
||
|
|
{
|
||
|
|
matrixc m_out = matrixc::Zeros(m.Rows(), m.Cols());
|
||
|
|
|
||
|
|
// Row-by-row matrix translation leveraging the underlying vector unpacker
|
||
|
|
for(ulong i = 0; i<m.Rows(); i++)
|
||
|
|
{
|
||
|
|
vector row = m.Row(i);
|
||
|
|
vectorc row_c = compact_to_complex(row,bp);
|
||
|
|
m_out.Row(row_c,i);
|
||
|
|
}
|
||
|
|
return m_out;
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|