Cond
Cond
Compute the condition number of a matrix.
double matrix::Cond(
const ENUM_MATRIX_NORM norm // matrix norm
);Parameters
- norm
[in] Order of the norm from ENUM_MATRIX_NORM
Return Value
The condition number of the matrix. May be infinite.
Note
The condition number of x is defined as the norm of x times the norm of the inverse of x [1]. The norm can be the usual L2-norm (root-of-sum-of-squares) or one of a number of other matrix norms.
The condition umber is the K value equal to the product of the matrix A norms and its inverse. Matrices with a high condition number are called ill-conditioned. Those with a low condition number are called well-conditioned. The inverse matrix is obtained using pseudo-inversion, so as not to be limited by the condition of squareness and non-singularity of the matrix.
An exception is the spectral condition number.
A simple algorithm for calculating the spectral condition number in MQL5:
double MatrixCondSpectral(matrix& a)
{
double norm=0.0;
vector v=a.Spectrum();
if(v.Size()>0)
{
double max_norm=v[0];
double min_norm=v[0];
for(ulong i=1; i<v.Size(); i++)
{
double real=MathAbs(v[i]);
if(max_norm<real)
max_norm=real;
if(min_norm>real)
min_norm=real;
}
max_norm=MathSqrt(max_norm);
min_norm=MathSqrt(min_norm);
if(min_norm>0.0)
norm=max_norm/min_norm;
}
return(norm);
}MQL5 example:
matrix a= {{1, 0, -1}, {0, 1, 0}, { 1, 0, 1}};
Print("a.Cond(MATRIX_NORM_P2)=", a.Cond(MATRIX_NORM_P2));
Print("a.Cond(MATRIX_NORM_FROBENIUS)=", a.Cond(MATRIX_NORM_FROBENIUS));
Print("a.Cond(MATRIX_NORM_INF)=", a.Cond(MATRIX_NORM_INF));
Print("a.Cond(MATRIX_NORM_MINUS_INF)=", a.Cond(MATRIX_NORM_MINUS_INF));
Print("a.Cond(MATRIX_NORM_P1)=)", a.Cond(MATRIX_NORM_P1));
Print("a.Cond(MATRIX_NORM_MINUS_P1)=", a.Cond(MATRIX_NORM_MINUS_P1));
Print("a.Cond(MATRIX_NORM_P2)=", a.Cond(MATRIX_NORM_P2));
Print("a.Cond(MATRIX_NORM_MINUS_P2)=", a.Cond(MATRIX_NORM_MINUS_P2));
/*
matrix a
[[1,0,-1]
[0,1,0]
[1,0,1]]
a.Cond(MATRIX_NORM_P2)=1.414213562373095
a.Cond(MATRIX_NORM_FROBENIUS)=3.162277660168379
a.Cond(MATRIX_NORM_INF)=2.0
a.Cond(MATRIX_NORM_MINUS_INF)=0.9999999999999997
a.Cond(MATRIX_NORM_P1)=)2.0
a.Cond(MATRIX_NORM_MINUS_P1)=0.9999999999999998
a.Cond(MATRIX_NORM_P2)=1.414213562373095
a.Cond(MATRIX_NORM_MINUS_P2)=0.7071067811865472
*/Python example:
import numpy as np
from numpy import linalg as LA
a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])
print("a \n",a)
print("LA.cond(a)=",LA.cond(a))
print("LA.cond(a, 'fro')=",LA.cond(a, 'fro'))
print("LA.cond(a, np.inf)=",LA.cond(a, np.inf))
print("LA.cond(a, -np.inf)=",LA.cond(a, -np.inf))
print("LA.cond(a, 1)=",LA.cond(a, 1))
print("LA.cond(a, -1)=",LA.cond(a, -1))
print("LA.cond(a, 2)=",LA.cond(a, 2))
print("LA.cond(a, -2)=",LA.cond(a, -2))
a
[[ 1 0 -1]
[ 0 1 0]
[ 1 0 1]]
LA.cond(a)= 1.4142135623730951
LA.cond(a, 'fro')= 3.1622776601683795
LA.cond(a, np.inf)= 2.0
LA.cond(a, -np.inf)= 1.0
LA.cond(a, 1)= 2.0
LA.cond(a, -1)= 1.0
LA.cond(a, 2)= 1.4142135623730951
LA.cond(a, -2)= 0.7071067811865475