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The first two steps of the Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.
There are examples where ... The Gram–Schmidt procedure gives a ... (1975), Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics, vol ...
This method has greater numerical stability than the Gram–Schmidt method above. The following table gives the number of operations in the k -th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size n .
A practical way to enforce this is by requiring that the next search direction be built out of the current residual and all previous search directions. The conjugation constraint is an orthonormal-type constraint and hence the algorithm can be viewed as an example of Gram-Schmidt orthonormalization. This gives the following expression:
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors.
Thus the problem of finding conjugate axes is less constrained than the problem of orthogonalization, so the Gram–Schmidt process works, with additional degrees of freedom that we can later use to pick the ones that would simplify the computation: Arbitrarily set .
An alternative way to arrive at the same expressions is to take the first three derivatives of the curve r′(t), r′′(t), r′′′(t), and to apply the Gram-Schmidt process. The resulting ordered orthonormal basis is precisely the TNB frame. This procedure also generalizes to produce Frenet frames in higher dimensions.
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form.When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: