jkalias
commented
1 year ago Hello, I'm testing the library to find solutions of systems of linear equations, and the result I get is always wrong. I give a couple of examples: The following system has no solution: x-y = 0 x + 3y = 4 3x + 5y = 9
However, I get as a solution x = 1.2083333333333335 y = 1.0416666666666667
which clearly violates the first equation x-y = 0. I have implemented it as follows: var A = Matrix.Build.DenseOfArray(new double[,] { { 1.0, -1.0 }, { 1.0, 3.0 }, { 3.0, 5.0 } }); var b = Vector.Build.Dense(new double[] { 0.0, 4.0, 9.0 }); var x = A.Solve(b);
It seems that the Solve method always returns approximate values, never the exact solution,
Hello. The solver is using a QR decomposition internally to solve the system. If the system is overdetermined as in your case, Q will be a 3x3 matrix, whereas R a 3x2 matrix, and the process of "solving" the system is essentially finding the x which minimizes ||Ax - b||. So, yes, not all equations can be satisfied. (see this link for more info).
I tried one of the examples in the documentation, which is the following:
var A = Matrix.Build.DenseOfArray(new double[,] { { 3, 2, -1 }, { 2, -2, 4 }, { -1, 0.5, -1 } }); var b = Vector.Build.Dense(new double[] { 1, -2, 0 }); var x = A.Solve(b);
And instead of getting the expected result x = -1, y = -2, z = -2, I get x = 1 y = -1.9999999999999996 z = -1.9999999999999992
I can't tell how you reached the conclusion that the expected solution is "x = -1, y = -2, z = -2", but it clearly isn't. Take the first equation for example:
3x + 2y - 1z = 3(-1)+2(-2)-1(-2) = -3-4+2 = -4 which is not equal to b[0] = 1
whereas for the Numerics solution = 3+2(-2)-1(-2) = 3-4+2 = 1
Hello, I'm testing the library to find solutions of systems of linear equations, and the result I get is always wrong. I give a couple of examples: The following system has no solution: x-y = 0 x + 3y = 4 3x + 5y = 9
However, I get as a solution x = 1.2083333333333335 y = 1.0416666666666667
which clearly violates the first equation x-y = 0. I have implemented it as follows: var A = Matrix.Build.DenseOfArray(new double[,] { { 1.0, -1.0 }, { 1.0, 3.0 }, { 3.0, 5.0 } });
var b = Vector.Build.Dense(new double[] { 0.0, 4.0, 9.0 });
var x = A.Solve(b);
It seems that the Solve method always returns approximate values, never the exact solution, because I tried one of the examples in the documentation, which is the following:
var A = Matrix.Build.DenseOfArray(new double[,] {
{ 3, 2, -1 },
{ 2, -2, 4 },
{ -1, 0.5, -1 }
});
var b = Vector.Build.Dense(new double[] { 1, -2, 0 });
var x = A.Solve(b);
And instead of getting the expected result x = -1, y = -2, z = -2, I get x = 1 y = -1.9999999999999996 z = -1.9999999999999992
Does anyone know what's the problem?