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Difference between revisions of "Orthogonalization method"

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The [[Классический метод ортогонализации|classical orthogonalization method for the QR decomposition of a square matrix (real version)]] is fairly simple. However, due to its instability, which manifests itself in the non-orthogonality of resulting systems, the method is very rarely used in practice.   
 
The [[Классический метод ортогонализации|classical orthogonalization method for the QR decomposition of a square matrix (real version)]] is fairly simple. However, due to its instability, which manifests itself in the non-orthogonality of resulting systems, the method is very rarely used in practice.   
  
Пусть имеются линейно независимые векторы <math>\mathbf{a}_1,\;\ldots,\;\mathbf{a}_N</math>.
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Let <math>\mathbf{a}_1,\;\ldots,\;\mathbf{a}_N</math> be linearly independent vectors. Define the projection of a vector <math>\mathbf{a}</math> on (the direction of) a vector <math>\mathbf{b}</math> by the formula <math>\mathbf{proj}_{\mathbf{b}}\,\mathbf{a} = {\langle \mathbf{a}, \mathbf{b} \rangle \over \langle \mathbf{b}, \mathbf{b}\rangle} \mathbf{b} ,</math>  
Пусть оператор проекции вектора <math>\mathbf{a}</math> на вектор <math>\mathbf{b}</math> определён следующим образом: <math>\mathbf{proj}_{\mathbf{b}}\,\mathbf{a} = {\langle \mathbf{a}, \mathbf{b} \rangle \over \langle \mathbf{b}, \mathbf{b}\rangle} \mathbf{b} ,</math>  
 
  
где <math>\langle \mathbf{a}, \mathbf{b} \rangle</math> — скалярное произведение векторов <math>\mathbf{a}</math> и <math>\mathbf{b}</math>.  
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where <math>\langle \mathbf{a}, \mathbf{b} \rangle</math> is the scalar product of the vectors <math>\mathbf{a}</math> and <math>\mathbf{b}</math>.  
  
Скалярное произведение для двух векторов  <math>\mathbf{ a= [a_1, a_2, ...,a_k]}</math> и <math>\mathbf{ b= [b_1, b_2, ..., b_k]}</math> в '''''k'''''-мерном действительном пространстве определяется как:
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In the '''''k'''''-dimensional real space, the scalar product of the vectors <math>\mathbf{ a= [a_1, a_2, ...,a_k]}</math> and <math>\mathbf{ b= [b_1, b_2, ..., b_k]}</math> is defined as
 
:<math>\langle \mathbf{a}, \mathbf{b} \rangle=\sum_{i=1}^k a_ib_i=a_1b_1+a_2b_2+\cdots+ a_kb_k</math>.
 
:<math>\langle \mathbf{a}, \mathbf{b} \rangle=\sum_{i=1}^k a_ib_i=a_1b_1+a_2b_2+\cdots+ a_kb_k</math>.
  

Revision as of 18:23, 5 March 2018


The basic authors of the description: Инжелевская Дарья Валерьевна(text), А.В.Фролов(editing)

The Gram--Schmidt orthogonalization is a method that constructs a set of orthogonal vectors {\displaystyle \mathbf {b}_{1},\;\ldots ,\;\mathbf {b} _{N}} or a set of orthonormal vectors {\displaystyle \mathbf {e} _{1},\;\ldots ,\;\mathbf {e}_{N}} from a given set of linearly independent vectors {\displaystyle \mathbf {a} _{1},\;\ldots ,\;\mathbf {a} _{N}}. This is done in such a way that each vector {\displaystyle \mathbf {b} _{j}} or {\displaystyle \mathbf {e} _{j}} is a linear combination of the vectors {\displaystyle \mathbf {a} _{1},\;\ldots ,\; \mathbf {a} _{j}}. The process may be used for obtaining the QR decomposition, where the system of original vectors is the columns of a given matrix, while the columns of Q are the result of orthogonalization. Thus, unlike the Givens (rotation) and Householder (reflection) methods, which are based on the left unitary/orthogonal reduction to triangular form, the orthogonalization method reduces the original matrix by right non-orthogonal (triangular) transformations to a unitary/orthogonal matrix.

Mathematical foundations of the method

The classical orthogonalization method for the QR decomposition of a square matrix (real version) is fairly simple. However, due to its instability, which manifests itself in the non-orthogonality of resulting systems, the method is very rarely used in practice.

Let \mathbf{a}_1,\;\ldots,\;\mathbf{a}_N be linearly independent vectors. Define the projection of a vector \mathbf{a} on (the direction of) a vector \mathbf{b} by the formula \mathbf{proj}_{\mathbf{b}}\,\mathbf{a} = {\langle \mathbf{a}, \mathbf{b} \rangle \over \langle \mathbf{b}, \mathbf{b}\rangle} \mathbf{b} ,

where \langle \mathbf{a}, \mathbf{b} \rangle is the scalar product of the vectors \mathbf{a} and \mathbf{b}.

In the k-dimensional real space, the scalar product of the vectors \mathbf{ a= [a_1, a_2, ...,a_k]} and \mathbf{ b= [b_1, b_2, ..., b_k]} is defined as

\langle \mathbf{a}, \mathbf{b} \rangle=\sum_{i=1}^k a_ib_i=a_1b_1+a_2b_2+\cdots+ a_kb_k.

Этот оператор проецирует вектор \mathbf{a} коллинеарно вектору \mathbf{b}.

Ортогональность векторов \mathbf{a} и \mathbf{b} достигается на шаге (2).

Классический процесс Грама — Шмидта выполняется следующим образом:

{\begin{array}{lclr} {\mathbf {b}}_{1}&=&{\mathbf {a}}_{1}&(1)\\ {\mathbf {b}}_{2}&=&{\mathbf {a}}_{2}-{\mathbf {proj}}_{{{\mathbf {b}}_{1}}}\,{\mathbf {a}}_{2}&(2)\\ {\mathbf {b}}_{3}&=&{\mathbf {a}}_{3}-{\mathbf {proj}}_{{{\mathbf {b}}_{1}}}\,{\mathbf {a}}_{3}-{\mathbf {proj}}_{{{\mathbf {b}}_{2}}}\,{\mathbf {a}}_{3}&(3)\\ {\mathbf {b}}_{4}&=&{\mathbf {a}}_{4}-{\mathbf {proj}}_{{{\mathbf {b}}_{1}}}\,{\mathbf {a}}_{4}-{\mathbf {proj}}_{{{\mathbf {b}}_{2}}}\,{\mathbf {a}}_{4}-{\mathbf {proj}}_{{{\mathbf {b}}_{3}}}\,{\mathbf {a}}_{4}&(4)\\ &\vdots &&\\{\mathbf {b}}_{N}&=&{\mathbf {a}}_{N}-\displaystyle \sum _{{j=1}}^{{N-1}}{\mathbf {proj}}_{{{\mathbf {b}}_{j}}}\,{\mathbf {a}}_{N}&(N) \end{array}}


На основе каждого вектора \mathbf{b}_j \;(j = 1 \ldots N) может быть получен нормированный вектор: \mathbf{e}_j = {\mathbf{b}_j\over \| \mathbf{b}_j \|} (у нормированного вектора направление будет таким же, как у исходного, а норма — единичной). Норма в формуле - согласованная со скалярным произведением: \| x \| = \sqrt{\langle x, x \rangle}

Результаты процесса Грама — Шмидта:

\mathbf{b}_1,\;\ldots,\;\mathbf{b}_N — система ортогональных векторов либо

\mathbf{e}_1,\;\ldots,\;\mathbf{e}_N — система ортонормированных векторов.

Наиболее используемой на практике формой метода является вариант метода ортогонализации с переортогонализацией.