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  2. Orthogonal Procrustes problem - Wikipedia

    en.wikipedia.org/wiki/Orthogonal_Procrustes_problem

    The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra.In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely maps to .

  3. Numerical methods for linear least squares - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows. = , where Q is an m×m orthogonal matrix (Q T Q=I) and R is an n×n upper triangular matrix with >. The residual vector is left-multiplied by Q T.

  4. Linear least squares - Wikipedia

    en.wikipedia.org/wiki/Linear_least_squares

    Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best ...

  5. Moore–Penrose inverse - Wikipedia

    en.wikipedia.org/wiki/Moore–Penrose_inverse

    A common use of the pseudoinverse is to compute a "best fit" (least squares) approximate solution to a system of linear equations that lacks an exact solution (see below under § Applications). Another use is to find the minimum norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement ...

  6. Regularized least squares - Wikipedia

    en.wikipedia.org/wiki/Regularized_least_squares

    This gives a more intuitive interpretation for why Tikhonov regularization leads to a unique solution to the least-squares problem: there are infinitely many vectors satisfying the constraints obtained from the data, but since we come to the problem with a prior belief that is normally distributed around the origin, we will end up choosing a ...

  7. Non-negative least squares - Wikipedia

    en.wikipedia.org/wiki/Non-negative_least_squares

    This algorithm takes a finite number of steps to reach a solution and smoothly improves its candidate solution as it goes (so it can find good approximate solutions when cut off at a reasonable number of iterations), but is very slow in practice, owing largely to the computation of the pseudoinverse ((A P) T A P) −1. [1]

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  9. Polynomial interpolation - Wikipedia

    en.wikipedia.org/wiki/Polynomial_interpolation

    The matrix X on the left is a Vandermonde matrix, whose determinant is known to be () = < (), which is non-zero since the nodes are all distinct. This ensures that the matrix is invertible and the equation has the unique solution A = X − 1 ⋅ Y {\displaystyle A=X^{-1}\cdot Y} ; that is, p ( x ) {\displaystyle p(x)} exists and is unique.