Orthogonal Projectors and Systems of Linear Algebraic Equations
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ogonal Projectors and Systems of Linear Algebraic Equations I. V. Kireev1, 2* 1
Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, Akademgorodok 50/44, Krasnoyarsk, 660036 Russia
2
Institute of Mathematics and Fundamental Informatics, Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660041 Russia Received February 19, 2018; in final form, January 31, 2020; accepted April 16, 2020
Abstract—In this paper, an operator iterative procedure for constructing an orthogonal projection of a vector onto a given subspace is proposed. The algorithm is based on Euclidean orthogonalization of power sequences of a special linear transform generated by an initial subspace. A numerical method based on this idea for solving consistent systems of linear algebraic equations is proposed. Numerical results are presented. DOI: 10.1134/S1995423920030064 Keywords: numerical methods, linear algebra, orthogonal projectors, Kaczmarz method, Krylov subspaces.
1. INTRODUCTION Let E denote a Euclidean space of dimension dim E with inner product (u, v) of vectors u, v ∈ E; u = (u, u) is the length of a vector u. Let V be a subspace of E, dim V < dim E; and let V⊥ be its orthogonal complement to E, that is, the annihilator [2] of the set V. Then E is a direct sum V ⊕ V⊥ and for any vector u ∈ E there exists a unique representation ⊥ ⊥ u = uV + u⊥ V : uV ∈ V, uV ∈ V ;
(1)
uV is called [2] the orthogonal projection, and u⊥ V is the orthogonal component of the vector u with respect to V. Let P and P ⊥ be the orthogonal projectors of the Euclidean space E onto V and V⊥ , respectively. Then ⊥ uV = Pu, u⊥ V = P u.
The problem of determining Pu for a given vector u is one of the key problems of computational linear algebra. In particular, solving a system of linear algebraic equations can be reduced to this problem. An iterative algorithm (in operator form) for constructing an orthogonal projection of a vector onto a subspace and a method based on this algorithm for numerically solving a consistent system of linear algebraic equations will be considered below. *
E-mail: [email protected]
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ORTHOGONAL PROJECTORS AND SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS
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2. CALCULATION OF THE ORTHOGONAL PROJECTION OF A VECTOR ONTO A SUBSPACE In what follows, any self-adjoint nonnegative definite transformation A of the subspace E whose range is V⊥ will be called an annihilating transformation of the subspace V. Thus, V coincides with the kernel of its annihilator A. Its condition number æA will be called the condition number of restriction of the operator A on V⊥ . With no rounding errors, the following theorem is valid: Theorem 1. For any annihilator A of a subspace V ⊂ E and any vector u ∈ / V consider the iterative process u0 := u, s1 := Au0 for n ≥ 1:
n
u := u
n−1
un−1 , sn n − s ; (sn , sn )
sn+1 := Aun −
(Aun , sn ) n s . (sn , sn )
(2)
In less than dim V⊥ steps, this will lead to a construction of an orthogonal projection Pu of the vector u onto the subspace V, that is, a natural number k ≤ dim
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