Normal Equations

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normal equations

[¦nȯr·məl i′kwā·zhənz]
The set of equations arising in the least squares method whose solutions give the constants that determine the shape of the estimated function.

Normal Equations


a set of algebraic or transcendental equations whose solution gives the approximate values of unknown quantities that can be derived by the method of least squares.

References in periodicals archive ?
9) corresponds to the normal equations of the matrix operator
It has been proven that the normal equations always have a unique solution provided f(x) [member of] C[a.
However, such a solution directly from normal equations is rather susceptible to round off errors and, more importantly, to the singularity of these equations (Madala & Ivakhnenko, 1994; Ivakhnenko & Muller, 1995ab; Atashkari et al.
Though CNF, CTF and PMCHWT formulations can provide satisfactory results in some particular cases such as those involving losses, the inaccuracy of the formulations employing only tangential or only normal equations in the matched case points out that the consideration of both types of equations is required.
The normal equation for computing [DELTA]y is a positive definite and symmetric matrix but the normal equation for computing [DELTA]x is not symmetric.
To derive the Normal equations for the multivariate model, we take partial derivatives of Q([b.
With the equation in this form, there are two more coefficients to be estimated than there are independent normal equations.
We derive the normal equations by least squares as in the section on simple linear regression, except that there are more--an additional one for each X variable added to the model.
However, if there is a large area without data or if the number of data points is small, then the coefficient matrix of the normal equations is not full rank and a unique solution cannot be obtained.
Methodology of evaluating a GOCO model was superposition of full normal equations plus optimal weighting of the mentioned models.
Our work to implement a direct solver for normal equations [8] is an extension of those efforts.