Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU . Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.

Keywords:
Toeplitz tridiagonal matrix; Crout’s method; tridiagonal and diagonally dominant matrix

Authorship

C. G. ALMEIDA ^**Corresponding author: César Guilherme Almeida - E-mail: cesargui@ufu.br

Núcleo de Matemática Aplicada da Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100, Brazil - E-mail: cesargui@ufu.brUniversidade Federal de UberlândiaBrazilBrazilNúcleo de Matemática Aplicada da Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100, Brazil - E-mail: cesargui@ufu.br

https://orcid.org/0000-0001-7619-8162

S. A. E. REMIGIO

Núcleo de Matemática Aplicada da Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100, Brazil - E-mail: santos.er@ufu.brUniversidade Federal de UberlândiaBrazilBrazilNúcleo de Matemática Aplicada da Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100, Brazil - E-mail: santos.er@ufu.br

https://orcid.org/0000-0002-5088-7311

*Corresponding author: César Guilherme Almeida - E-mail: cesargui@ufu.br

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (35)

(2.1)

(2.2)

(3.1)

(2.1)

A = [\begin{matrix} d_{1} & a_{1} & 0 & \dots & 0 \\ b_{2} & d_{2} & a_{2} & ⋮ \\ 0 & b_{3} & d_{3} & a_{3} \\ ⋱ & ⋱ & ⋱ \\ ⋮ \\ b_{n - 1} & d_{n - 1} & a_{n - 1} \\ 0 & \dots & b_{n} & d_{n} \end{matrix}] .

(2.2)

L = [\begin{matrix} α_{1} & 0 & 0 & \dots & 0 \\ b_{2} & α_{2} & 0 & ⋮ \\ 0 & b_{3} & α_{3} \\ ⋱ & ⋱ & ⋱ \\ ⋮ & b_{n - 1} & α_{n - 1} & 0 \\ 0 & \dots & b_{n} & α_{n} \end{matrix}], U = [\begin{matrix} 1 & \frac{a_{1}}{α_{1}} & 0 & \dots & 0 \\ 0 & 1 & \frac{a_{2}}{α_{2}} & ⋮ \\ 0 & 1 & \frac{a_{3}}{α_{3}} \\ ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & 0 & 1 & \frac{a_{n - 1}}{α_{n - 1}} \\ 0 & \dots & 0 & 1 \end{matrix}],

(3.1)

M_{k} = d M_{k - 1} - a^{2} M_{k - 2}, \forall k > 2 .

M_{2 k} (- d) = - d M_{2 k - 1} (- d) - a^{2} M_{2 k - 2} (- d) = d M_{2 k - 1} (d) - a^{2} M_{2 (k - 1)} (d) = M_{2 k} (d) .

\lim_{d \to 0} M_{2 m - 1} (- d) = - \lim_{d \to 0} M_{2 m - 1} (d) = - M_{2 m - 1} (0)

\lim_{d \to 0} M_{2 m - 1} (- d) = M_{2 m - 1} (- \lim_{d \to 0} d) = M_{2 m - 1} (0) .

(i) M_{2 k} (0) = (- 1)^{k} | a |^{2 k}; (i i) {[\frac{1}{d} M_{2 k + 1} (d)]}_{∣ d = 0} = (- 1)^{k} | a |^{2 k} (k + 1) .

M_{2 k} (d) = d M_{2 k - 1} (d) - a^{2} M_{2 k - 2} (d) .

\frac{1}{d} M_{3} (d) = M_{2} (d) - a^{2} . Thus, {[\frac{1}{d} M_{3} (d)]}_{∣ d = 0} = (- 1)^{1} | a |^{2} - | a |^{2} = (- 1)^{1} 2 | a |^{2} .

{[\frac{1}{d} M_{2 m + 1} (d)]}_{∣ d = 0} = (- 1)^{m} | a |^{2 m} (m + 1), \forall m, 1 \leq m < k .

{[\frac{1}{d} M_{2 k + 1} (d)]}_{∣ d = 0} = M_{2 k} (0) - | a |^{2} {[\frac{1}{d} M_{2 k - 1} (d)]}_{∣ d = 0} .

{[\frac{1}{d} M_{2 k + 1} (d)]}_{∣ d = 0} = (- 1)^{k} | a |^{2 k} - | a |^{2} (- 1)^{k - 1} | a |^{2 (k - 1)} (k - 1 + 1) = (- 1)^{k} | a |^{2 k} (k + 1) .

M_{k} (| a | x) = | a | x M_{k - 1} (| a | x) - a^{2} M_{k - 2} (| a | x), \forall k > 2 .

(i) p_{2 k} (0) = (- 1)^{k}; (i i) {[\frac{1}{x} p_{2 k + 1} (x)]}_{∣ x = 0} = (- 1)^{k} (k + 1) .

(i) p_{2 k} (x) = \frac{1}{| a |^{2 k}} M_{2 k} (| a | x) \to p_{2 k} (0) = \frac{1}{| a |^{2 k}} M_{2 k} (0); (ii) \frac{1}{x} p_{2 k + 1} (x) = \frac{1}{x} \frac{1}{| a |^{2 k + 1}} M_{2 k + 1} (| a | x) = \frac{1}{| a |^{2 k}} [\frac{1}{| a | x} M_{2 k + 1} (| a | x)] \to {[\frac{1}{x} p_{2 k + 1} (x)]}_{∣ x = 0} = \frac{1}{| a |^{2 k}} {[\frac{1}{d} M_{2 k + 1} (d)]}_{∣ d = 0}, where d = | a | x . □

M_{k} (2 | a |) = 2 | a | [(2 k - 3) | a |^{k - 1}] - | a |^{2} [(2 k - 5) | a |^{k - 2}] = (2 k - 1) | a |^{k} .

(i) p_{k} (x) = (x^{2} - x_{1}^{2}) \dots (x^{2} - x_{l}^{2}), if k is an even number; (ii) p_{k} (x) = x (x^{2} - x_{1}^{2}) \dots (x^{2} - x_{l}^{2}), if k is an odd number.

x_{2 m}^{(1)} \in (0, x_{2 m - 1}^{(1)}), x_{2 m}^{(i)} \in (x_{2 m - 1}^{(i - 1)}, x_{2 m - 1}^{(i)}), 2 \leq i \leq m - 1, x_{2 m}^{(m)} \in (x_{2 m - 1}^{(m - 1)}, 2), \forall m, m \geq 2; x_{2 m + 1}^{(j)} \in (x_{2 m}^{(j)}, x_{2 m}^{(j + 1)}), 1 \leq j \leq m - 1, x_{2 m + 1}^{(m)} \in (x_{2 m}^{(m)}, 2), \forall m, m \geq 1 .

x_{2}^{(1)} = 1; x_{3}^{(1)} = \sqrt{2}; x_{4}^{(1)} = \sqrt{\frac{3 - \sqrt{5}}{2}} and x_{4}^{(2)} = \sqrt{\frac{3 + \sqrt{5}}{2}}; x_{5}^{(1)} = 1 and x_{5}^{(2)} = \sqrt{3} .

x_{3}^{(1)} \in (x_{2}^{(1)}, 2); x_{4}^{(1)} \in (0, x_{3}^{(1)}) and x_{4}^{(2)} \in (x_{3}^{(1)}, 2); x_{5}^{(1)} \in (x_{4}^{(1)}, x_{4}^{(2)}) and x_{5}^{(2)} \in (x_{4}^{(2)}, 2) .

(I) x_{2 k}^{(1)} \in (0, x_{2 k - 1}^{(1)}), (II) x_{2 k}^{(i)} \in (x_{2 k - 1}^{(i - 1)}, x_{2 k - 1}^{(i)}), 2 \leq i \leq k - 1,

(IV) x_{2 k + 1}^{(j)} \in (x_{2 k}^{(j)}, x_{2 k}^{(j + 1)}), 1 \leq j \leq k - 1, and (V) x_{2 k + 1}^{(k)} \in (x_{2 k}^{(k)}, 2) .

p_{2 k} (x_{2 k - 1}^{(1)}) = - p_{2 k - 2} (x_{2 k - 1}^{(1)}),

s g n (p_{2 k} (x_{2 k - 1}^{(1)})) = - (- 1)^{k - 2} = (- 1)^{k - 1} .

p_{2 k} (0) p_{2 k} (x_{2 k - 1}^{(1)}) < 0 .

p_{2 k} (x_{2 k - 1}^{(i - 1)}) = - p_{2 k - 2} (x_{2 k - 1}^{(i - 1)}), p_{2 k} (x_{2 k - 1}^{(i)}) = - p_{2 k - 2} (x_{2 k - 1}^{(i)}) .

x_{2 k - 1}^{(i - 1)} \in (x_{2 k - 2}^{(i - 1)}, x_{2 k - 2}^{(i)}), 2 \leq i \leq k - 1, x_{2 k - 1}^{(i)} \in (x_{2 k - 2}^{(i)}, x_{2 k - 2}^{(i + 1)}), 1 \leq i \leq k - 2 .

s g n (p_{2 k} (x_{2 k - 1}^{(i - 1)})) = - (- 1)^{k - i} = (- 1)^{k - i + 1}, s g n (p_{2 k} (x_{2 k - 1}^{(i)})) = - (- 1)^{k - i - 1} = (- 1)^{k - i} .

s g n (p_{2 k} (x_{2 k - 1}^{(i - 1)}) p_{2 k} (x_{2 k - 1}^{(i)})) = (- 1)^{k - i + 1} (- 1)^{k - i} = (- 1)^{2 (k - i) + 1} = - 1 .

p_{2 k} (x_{2 k - 1}^{(k - 1)}) = - p_{2 k - 2} (x_{2 k - 1}^{(k - 1)}),

p_{2 k + 1} (x_{2 k}^{(j)}) = - p_{2 k - 1} (x_{2 k}^{(j)}) and p_{2 k + 1} (x_{2 k}^{(j + 1)}) = - p_{2 k - 1} (x_{2 k}^{(j + 1)}) .

x_{2 k}^{(j)} \in (x_{2 k - 1}^{(j - 1)}, x_{2 k - 1}^{(j)}), 2 \leq j \leq k - 1, x_{2 k}^{(j + 1)} \in (x_{2 k - 1}^{(j)}, x_{2 k - 1}^{(j + 1)}), 1 \leq j \leq k - 2 .

s g n (p_{2 k + 1} (x_{2 k}^{(j)})) = - (- 1)^{k - j} = (- 1)^{k - j + 1}, s g n (p_{2 k + 1} (x_{2 k}^{(j + 1)})) = - (- 1)^{k - j - 1} = (- 1)^{k - j} .

s g n (p_{2 k + 1} (x_{2 k}^{(j)}) p_{2 k + 1} (x_{2 k}^{(j + 1)})) = (- 1)^{k - j + 1} (- 1)^{k - j} = (- 1)^{2 (k - j) + 1} = - 1 .

p_{2 k + 1} (x_{2 k}^{(k)}) = - p_{2 k - 1} (x_{2 k}^{(k)}),

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Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

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[1] *Corresponding author: César Guilherme Almeida - E-mail: cesargui@ufu.br