Solving equations with the aid of inverse interpolation spline functions

9 years ago

Abstract

We consider the solving of a nonlinear equation in $\mathbb{R}$. We construct a spline function which approximates the nonlinear mapping, and we use the Hermite polynomial with two nodes (one simple and one multiple), to construct a iterative method.

Authors

Crăciun Iancu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Title

Original title (in French)

Resolution des equations à l’aide des fonctions splines d’interpolation invèrse

English translation of the title

Solving equations with the aid of inverse interpolation spline functions

Keywords

Hermite interpolation; inverse interpolation; spline functions; iterative methods; nonlinear equations in R

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Cite this paper as:

C. Iancu, I. Păvăloiu, Resolution des equations à l’aide des fonctions splines d’interpolation invèrse, Babes-Bolyai University, Faculty of Mathematics, Seminar on functional analysis and numerical methods, Preprint no. 1 (1984), pp. 97-104 (in French).

About this paper

Journal

Seminar on functional analysis and numerical methods,
Preprint

Publisher Name

“Babes-Bolyai” University,
Faculty of Mathematics and Physics,
Research Seminars

DOI

Not available yet.

References

[1] C. Iancu, Analiza si prelucrarea datelor cu ajutorul functiilor spline. Teza de doctorat, Cluj (1983), Facultatea de Matematica a Univ. ”Babes-Bolyai”.

[2] A. Imamov, Resenie nelineinib uravnenii metodom obratnogo splaininterpolirovaniis. Metodi splain-funktii, Akademia Nauk SSSR, Novosibirsk, 81 (1979), 74–80

[3] I. Pavaloiu, Rezolvarea ecuatiilor prin interpolare. Ed. Dacia, Cluj, 1981.

[4] Turowicz, B.A., Sur les derivees d’ordre superieur d’une fonction inverse, Colloq. Math. (1959), 83–87.

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"Babeş-Bolyai" University

Faculty of Mathematics and Physics

Research Seminars

Seminar on Functional Analysis and Numerical Methods

Preprint Nr.1, 1984, pp. 97-104

Solving equations with the aid of inverse interpolation spline functions

by
C. Iancu and I. Păvăloiu

In the following we present a generalization of the result presented in the work [ 2 ] , concerning the resolution of equations using inverse interpolation spline functions.

Consider the equation

(1)

f (x) = 0

Or $f : I \to ℝ$ is a real function of a real variable and I is an interval of the real axis.

We designate by $AND$ a set of $m$ real numbers distinct from I, in particular

(2)

x_{1} < x_{2} < \dots < x_{m}

Regarding the function $f$ we will assume that its values are known ${and}_{i}^{(0)} = f (x_{i}), i = 1, 2, \dots, m$ as well as the values of its successive derivatives up to the order $n - 1$ to the point $x_{1}, n \in ℕ$ , that's to say $f^{'} (x_{1}) = {and}_{1}^{(1)}, f^{''} (x_{1}) = {and}_{1}^{(2)}, \dots, f^{(n - 1)} (x_{1}) = {and}_{1}^{(n - 1)}$ .

Such a function can be obtained as a result of an experiment or, for example, as a numerical solution to a Cauchy-type problem relating to a differential equation.

To fix the ideas, we will assume that equation ( 1 ) admits a single root $\bar{x} \in I$ and that there are two real numbers $x_{p}, x_{p + 1} \in AND$ such as $f (x_{p}) f (x_{p + 1}) < 0,$ that's to say $x \in (x_{p}, x_{p + 1})$ .

In the work [ 2 ] the author constructs inverse interpolation spline functions of the third degree, with the help of which he proceeds to the approximation of the roots of the equations of the form ( 1 ).

In the following we propose to use the two-node Hermite-type inverse interpolation polynomial, studied in [ 3 ] , in order to present a generalization of the results contained in [ 2 ] .

We designate by ${In}_{x_{1}}$ a neighborhood of the point $x_{1}$ and write $F_{1} = f ({In}_{x_{1}})$ . We will subsequently assume that the restriction of the function $f$ to the whole ${In}_{x_{1}}$ is bijective and that ${and}_{1}^{(1)} \neq 0;$ in this case, the successive derivatives of the function $f^{- 1}$ to the point ${and}_{1}$ can be obtained using the formula [ 4 ] :

(3)

{[f^{- 1} ({and}_{1})]}^{(k)} = \sum \frac{(2 k - 2 - i_{1})! {(- 1)}^{k - 1 + i_{1}}}{i_{2}! \dots i_{k}! {(f^{'} (x_{1}))}^{2 k - 1}} {(\frac{f^{'} (x_{1})}{1!})}^{i_{1}} \dots {(\frac{f^{(k)} (x_{1})}{k!})}^{i_{k}}

where the above sum is over all integer and non-negative solutions of the system of equations

(4)		$i_{2} + 2 i_{3} + \dots + (k - 1) i_{k}$	$= k - 1$
	$i_{1} + i_{2} + \dots + i_{k}$	$= k - 1, k = 1, 2, \dots, n - 1 .$

The inverse interpolation Hermite polynomial at points ${and}_{1}^{(0)}, {and}_{2}^{(0)}$ will then take the following form:

(5)

P_{1} (and)

= \sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - j - 1} {[f^{- 1} ({and}_{1})]}^{(j)} \frac{1}{k! j!} {[\frac{{(and - {and}_{1})}^{n}}{{oh}_{1} (and)}]}_{and = {and}_{1}}^{(k)} \frac{{oh}_{1} (and)}{{(and - {and}_{1})}^{n - j - k}} + x_{2} {(\frac{and - {and}_{1}}{{and}_{2} - {and}_{1}})}^{n}

(6)

{oh}_{1} (and) = {(and - {and}_{1})}^{n} (and - {and}_{2})

We will designate by $P_{s} (and)$ the inverse Hermite interpolation polynomial in the interval $[{and}_{s}, {and}_{s + 1}]$ who meets the conditions:

(7)		$P_{s}^{(r)} ({and}_{s})$	$= P_{s - 1}^{(r)} ({and}_{s}), r = 0, 1, \dots, n - 1$
	$P_{s} ({and}_{s + 1})$	$= x_{s + 1}$

Under these conditions, $P_{s} (and)$ will take the following form

(8)			$P_{s} (and) =$
		$= \sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - j - 1} {[P_{s - 1} ({and}_{s})]}^{(j)} \frac{1}{k! j!} {[\frac{{(and - {and}_{s})}^{n}}{{oh}_{s} (and)}]}_{and = {and}_{n}}^{(k)} \frac{{oh}_{s} (and)}{{(and - {and}_{s})}^{n - j - k}} + x_{s + 1} {(\frac{and - {and}_{s}}{{and}_{s + 1} - {and}_{s}})}^{n},$

(9)

{oh}_{s} (and) = {(and - {and}_{s})}^{n} (and - {and}_{s + 1})

For $s = 2, 3, \dots, p$ .

It is easy to see that the expression ( 8 ) can be written in the form

(10)

P_{s} (and)

= \sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - j - 1} {[P_{s - 1} ({and}_{s})]}^{(j)} \frac{{(- 1)}^{k} {(and - {and}_{s})}^{j + k} (and - {and}_{s + 1})}{j! {({and}_{s} - {and}_{s + 1})}^{k + 1}} + x_{s + 1} {(\frac{and - {and}_{s}}{{and}_{s + 1} - {and}_{s}})}^{n}

Or $s = 2, 3, \dots, p$ .

An approximate value for the root $\bar{x}$ of equation ( 1 ) is given by

(11)

\bar{x} ≃ P_{p} (0),

that's to say

(12)

\bar{x} ≃ \sum_{j = 0}^{n - 1} \sum_{k = 0}^{n - j - 1} {[P_{p - 1} ({and}_{p})]}^{(j)} \frac{{(- 1)}^{j + 1} {and}_{p}^{j + k} {and}_{p + 1}}{j! {({and}_{p} - {and}_{p + 1})}^{k + 1}} + \frac{{(- 1)}^{n} x_{p + 1} {and}_{p}^{n}}{{({and}_{p + 1} - {and}_{p})}^{n}}

We will now deal with two special cases of the problem presented above.

1. The case $n = 2$ . In this case, the polynomials ( 10 ) take the following form

(13)		$P_{s} (and)$	$= \sum_{j = 0}^{1} \sum_{k = 0}^{1 - j} {[P_{s - 1} ({and}_{s})]}^{(j)} \frac{{(- 1)}^{k} {(and - {and}_{s})}^{j + k} (and - {and}_{s + 1})}{j! {({and}_{s} - {and}_{s + 1})}^{k + 1}} + x_{s + 1} {(\frac{and - {and}_{s}}{{and}_{s + 1} - {and}_{s}})}^{2},$
	$s$	$= 2, 3, \dots, p$

And

(14)

P_{1} (and)

= \sum_{j = 0}^{1} \sum_{k = 0}^{1 - j} {[f^{- 1} ({and}_{1})]}^{(j)} \frac{{(- 1)}^{k} {(and - {and}_{1})}^{j + k} (and - {and}_{2})}{j! {({and}_{1} - {and}_{2})}^{k + 1}} + x_{2} {(\frac{and - {and}_{1}}{{and}_{2} - {and}_{1}})}^{2}

It is easy to see that the expression ( 12 ) can be put in this case in the form [ 1 ]

(15)

\bar{x} ≃ x_{p} + a_{p} {and}_{p}^{2} - {and}_{p} P_{p - 1}^{'} ({and}_{p})

(16)

a_{p} = \frac{1 - P_{p - 1}^{'} ({and}_{p}) [x_{p}, x_{p + 1}; f]}{(x_{p + 1} - x_{p}) {[x_{p}, x_{p + 1}; f]}^{2}}

2. The case $n = 3$ . In this case ( 10 ) is written

(17)

P_{s} (and)

= \sum_{j = 0}^{2} \sum_{k = 0}^{2 - j} {[P_{s - 1} ({and}_{s})]}^{(j)} \frac{{(- 1)}^{k} {(and - {and}_{s})}^{j + k} (and - {and}_{s + 1})}{j! {({and}_{s} - {and}_{s + 1})}^{k + 1}} + x_{s + 1} {(\frac{and - {and}_{s}}{{and}_{s + 1} - {and}_{s}})}^{3}

with

(18)

P_{1} (and)

= \sum_{j = 0}^{2} \sum_{k = 0}^{2 - j} {[f^{- 1} ({and}_{1})]}^{(j)} \frac{{(- 1)}^{k} {(and - {and}_{1})}^{j + k} (and - {and}_{2})}{j! {({and}_{1} - {and}_{2})}^{k + 1}} + x_{2} {(\frac{and - {and}_{1}}{{and}_{2} - {and}_{1}})}^{3}

and ( 12 ) is written [ 1 ]

(19)

\bar{x} ≃ x_{p} - P_{p - 1}^{'} ({and}_{p}) {and}_{p} + \frac{P_{p - 1}^{''} ({and}_{p})}{2} {and}_{p}^{2} - a_{p} {and}_{p}^{3}

(20)

a_{p} = \frac{1 - P_{p - 1}^{'} ({and}_{p}) [x_{p}, x_{p + 1}; f] - \frac{P_{p - 1}^{''} ({and}_{p})}{2} {[x_{p}, x_{p + 1}; f]}^{2} (x_{p + 1} - x_{p})}{{[x_{p}, x_{p + 1}; f]}^{3} (x_{p + 1} - x_{p})}

Numerical example.

We consider the equation

(21)

f (x) = 4 x^{3} + 3 x^{2} + 3 x - 1 = 0

which admits the only real root $x = 0, 25 .$

We will assume that with respect to the function $f$ from ( 21 ) we know the following values:

(22)

{\begin{matrix} f (0) & = - 1 \\ f^{'} (0) & = 3 \\ f^{''} (0) & = 6 \\ f (0.1) & = - 0.666 \\ f (0.2) & = - 0.248 \\ f (0.3) & = 0.278 \end{matrix}

It follows from ( 22 ) that $x_{p} = 0.2$ And $x_{p + 1} = 0.3 .$

If we use the string method only once in the interval $(0.2; 0.3)$ we get for $\bar{x}$ the following approximate value

\bar{x} ≃ 0.2471483

Applying the method given by ( 15 ) we obtain for $\bar{x}$ the approximate value

\bar{x} ≃ 0.2501480

while method ( 19 ) leads us to the following approximate value

\bar{x} ≃ 0.2504372

We note that in the case of the example treated the method which gives the best approximation of the root of the equation ( 21 ) is the method of inverse interpolation with the second degree spline function.

In the approximation formula given by ( 15 ) for the root $\bar{x}$ of equation ( 1 ), obtained using the second-order inverse interpolation spline function, is the value of the derivative $P_{p - 1}^{'} ({and}_{p})$ of the polynomial $P_{p - 1}$ to the point ${and}_{p}$ .

It is easily seen that this value can be obtained using the first-order divided differences of the function $f$ taken on consecutive nodes and using $f^{'} (x_{1})$ .

$P_{p - 1}^{'} ({and}_{p})$ which appears in ( 15 ) is expressed in particular using the following algorithm:

(23)

{\begin{matrix} P_{1}^{'} ({and}_{2}) = \frac{2}{[x_{1}, x_{2}; f]} - \frac{1}{f^{'} (x_{1})} \\ P_{p - 1}^{'} ({and}_{p}) = \frac{2}{[x_{p - 1}, x_{p}; f]} + 2 \sum_{k = 2}^{p - 1} \frac{{(- 1)}^{k}}{[x_{k - 1}, x_{k}; f]} - \frac{1}{f^{'} (x_{1})}, \end{matrix}

and $p$ is an even natural number, or

(24)

P_{p - 1}^{'} ({and}_{p})

= \frac{2}{[x_{p - 1}, x_{p}; f]} + 2 \sum_{k = 2}^{p - 1} \frac{{(- 1)}^{k + 1}}{[x_{k - 1}, x_{k}; f]} + \frac{1}{f^{'} (x_{1})}

and $p$ is an odd natural number, where

[x_{i}, x_{i + 1}; f] = \frac{{and}_{i + 1} - {and}_{i}}{x_{i + 1} - x_{i}}, i = 1, 2, \dots, p - 1 .

It is difficult to obtain formulas analogous to those given by ( 23 ) and ( 24 ) for the calculation of the values of the successive derivatives of the polynomial $P_{p - 1}$ to the point ${and}_{p}$ in the general case and even if this can be done, they take a very complicated form.

Bibliography

[1] C. Iancu, Data analysis and processing using spline functions . Doctoral thesis, Cluj (1983), Faculty of Mathematics of Babeş-Bolyai University.
[2] A. Imamov, Reşenie nelineinîb uravnenii metodom obratnogo splain-interpolovaniis. Metodî splain-funkţii , Akademia Nauk SSSR, Novosibirsk, 81 (1979), 74–80
[3] I. Pavaloiu, ^†^†margin: $\leftarrow$ clickable Solving equations by interpolation . Dacia Publishing House, Cluj, 1981.
[4] Turowicz, BA, On higher-order derivatives of an inverse function , Colloq. Math. (1959), 83–87.