In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model. The resulting fitted model can be used to summarize the data, to predict unobserved values from the same system, and to understand the mechanisms that may underlie the system.
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QR decomposition
In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm. If A has linearly independent columns (say n columns), then the first n columns of Q form an orthonormal basis for the column space of A.
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MATLAB
MATLAB (matrix laboratory) is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, Java, and Fortran.
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Iterative refinement
Iterative refinement is an iterative method proposed by James H. Wilkinson to improve the accuracy of numerical solutions to systems of linear equations. When solving a linear system Ax = b, due to the presence of rounding errors, the computed solution x̂ may sometimes deviate from the exact solution x. Starting with x1 = x̂, iterative refinement computes a sequence {x1,x2,x3,…} which converges to x when certain assumptions are met.
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Iterative method
In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations.
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Frontal solver
A frontal solver, due to Irons is an approach to solving sparse linear systems which is used extensively in finite element analysis. It is a variant of Gauss elimination that automatically avoids a large number of operations involving zero terms. A frontal solver builds a LU or Cholesky decomposition of a sparse matrix given as the assembly of element matrices by assembling the matrix and eliminating equations only on a subset of elements at a time.
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Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. More generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that vector space will be the elements of the associated field.
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Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Where an exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, or common sense.
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