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DADiSP Worksheet Functions > Function Categories > Matrix Math and Special Matrices > ^ (Matrix Solve)
Solves for a matrix.
A \^ b
|
A |
- |
A non-singular, usually square matrix. |
|
b |
- |
A matrix |
A matrix that produces b when multiplied by A.
W1: {{1, 4, 7},
{2, 5, 8},
{3, 6, 0}}
W2: {1,
2,
3}
W3: W1 *^ W2
W4: W1 \^ W3
W3 and W4 contain the following matrices:
W3 == {30,
36,
15}
W4 == {1,
2,
3}
Specifically, W4 solves the following system of equations:
x + 4y + 7z = 30
2x + 5y + 8z = 36
3x + 6y = 15
x == 1
y == 2
z == 3
Consider the following over-determined system of equations:
x + 4y + 7z = 30
2x + 5y + 8z = 36
3x + 6y = 15
x + 2y + z = 2
A = {{1, 4, 7},
{2, 5, 8},
{3, 6, 0},
{1, 2, 1}}
x = {30,
36,
15,
2}
b = A \^ x;
b == {-1.8,
3.2,
2.8}
y = A *^ b;
y == {30.6,
34.8,
13.8,
7.4}
norm(x-y) == 5.6921
The \^ operator finds a solution y that minimizes the mean squared error. See LINFIT to fit an arbitrary set of basis functions to a series using the method of least squares.
If A, b, and x are matrices, such that A *^ x = b, then A \^ b returns the matrix x.
A \^ b is equivalent to mdiv(A, b).
For A \^ b, where A is square, the system is solved using LU decomposition. This is usually numerically more stable than directly calculating the inverse matrix, i.e.
x = inv(A) *^ b.
If matrix A is not square, the system is considered a least squares problem and is solved by QR decomposition. The resulting matrix is the best solution in the least squares sense.
For scalars, a \^ b == a \ b == b * (1/a) == b / a. For example:
10 \^ 2 == 10 \ 2 == 2 * (1/10) == 0.2
See DADiSP/MatrixXL to significantly optimize \^.