A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
APPROXIMATIONS; LINEAR PROGRAMMING;NON- LINEAR FUNCTIONS; PROJECT MANAGEMENT WITH PERT/CPM; DECISION THEORY; THEORY OF GAMES; INVENTORY MODELLING; QUEUING THEORY
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the method has at most an order of convergence (at least) linear. Keywords: Iteration Function, Order of Convergence, Fractional Derivative.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
1. Roots of equations
Is y=f(x). The values of x that make y= 0 are called roots of the equation. The fundamental
theorem of algebra states that every polynomial of degree n has n roots. In the case of
real estate, it must correspond to x values that make the feature cut the x-axis:
The roots of a polynomial can be real or complex. If a polynomial has coefficients a0, a1, a2,
…an-1, an, real, then all the complex roots always occur in complex conjugate pairs. For
example, a cubic polynomial has the following general form:
f(x)= a0x3+ a1x2+ a2x+ a3
The fundamental theorem of algebra states that a polynomial of degree n has n roots. In
the case of cubic polynomial can be the following:
· Three distinct real roots.
· A real root with multiplicity 3.
· A simple real root and one real root with multiplicity 2.
· One real root and a complex conjugate pair.
Example. The roots of these polynomials are summarized below.
1. Three distinct real roots:
f1(x) = x3- 3x2- x + 3
=(x-3)(x-3)(x+1)
2. For their study, the functions can be classified into algebraic and transcendental.
Algebraic functions
Let g = f (x) the function expressed
fnyn+ fn-1yn-1+…+ f1y+ f0=0
Where fi is a polynomial of order i in x. Polynomials are a simple case of algebraic
functions that are usually represented as
fn(x)=a0+a1x+ a2x2…+ anxn
Where n is the order of the polynomial.
Example.
f2(x) = 1-2.37 x + 7.5 x 2
f6(x) = 5 x 2-x 3 + 7 x 6
Transcendental functions
Are those that are not algebraic. They include the trigonometric, exponential, logarithmic,
among others.
Example.
f(x) = lnx2-1
f(x) = e-0.2xsin(3x-5)
The methods described in this unit require that the function is differentiable in the range
where they apply. If the methods used in non-differentiable or discontinuous functions at
some points, to reach the result will depend, at random, that during the implementation
of the method do not touch those points.
On the other hand, the roots of the equations can be real or complex.
Real roots of algebraic and transcendental equations
In general, the methods for finding the real roots of algebraic equations and
transcendental methods are divided into intervals and open methods.
The interval methods exploit the fact that typically a function changes sign in the vicinity
of a root. They get this name because it needs two initial values to be "encapsulated" to
the root. Through such methods will gradually reduce the size of the interval so that the
3. repeated application of the methods always produce increasingly close approximations to
the actual value of the root, so methods are said to be convergent.
The open methods, in contrast, are based on formulas that require a single initial
value x (Initial approach to the root). Sometimes these methods away from the real value
of the root grow the number of iterations, i.e. diverge.
Bisection method
This method, also known as range partitioning method, part of an algebraic or
transcendental equation f (x) and an interval [x 1, x 2] such that f (x 1) and f (x 2) have
opposite signs , ie such that there is at least one root in that interval.
Once the interval [x 1, x 2] and secured the continuity of function within this range, it is
evaluated at the midpoint x m the interval. If f (x m) and f (x 1) have opposite signs, will
reduce the range of x 1 to x m, and that within these values is the desired
root. If f (x m) and f (x 1) have the same sign, will reduce the range of x m to x 2. By repeating
this process to make the difference between the two values of x m is a good approximation
of the root.
The algorithm of the method is as follows:
1. Choosing values x 1 and x 2 of the interval.
2. Checking the existence of a root in the interval [x 1, x 2] making sure
f(x1)f(x2)<0. Otherwise, it will be necessary to choose other values for x 1 and x 2.
3. Take and calculate f(xm) .
4. If f (xm)= 0 is found the root of the function (end of method). Otherwise, go to step
5.
5. Let T the desired tolerance (the margin of error). If <T was found closer
to the root with a margin of error of less than T (end method). Otherwise, go to
step 6.
6. If f(x1)f(x2) , Then do and repeat from 3, otherwise, make and
repeat from 3.
4. Fixed Point Method
Starting from an initial value directed vertically to the curve y = g (x) thereof,
horizontally to the line y = x, and again the curve vertically, horizontally to the line, etc.
The algorithm of the method is as follows:
1. Choose an initial approximation x0 .
2. Calculate g(x0 and make x0=g(x0).
Let n = 1.
3. Calculate xn=g(xn-1).
4. Compare | xn+1- xn| with | xn- xn-1| :
a) If | xn+1- xn|<| xn- xn-1|. The method converges. Go to step 5.
b) If | xn+1- xn|>| xn- xn-1|. The method diverges. He stops the method and
choosing a new approach x0.
c) If |xn+1- xn|=| xn- xn-1|. The method has stalled. He stops the method and
choosing a new approach x0.
Note that the first iteration is not yet possible to apply this criterion, so you omit
this step and continues in May.
5. If | xn- xn-1| = 0, found the root of the function (end method). Otherwise, go to
step 6.
6. Let T the desired tolerance (the margin of error). If | xn- xn-1| <T was found
closer to the root with a margin of error of less than T (end method). Otherwise, go
to step 3 by n = n +1.
5. Newton-Raphson Method
This method is part of a first
approximation xn and by applying a
recursive formula is close to the root of
the equation, so that new
approach xn+1 is located at the
intersection of the tangent to the curve
of the function at the point xn and the
x-axis.
The Newton-Raphson method uses an iterative process to approach one root of a
function. The specific root that the process locates depends on the initial, arbitrarily
chosen x-value.
Here, xn is the current known x-value, f(xn) represents the value of the function at xn, and
f'(xn) is the derivative (slope) at xn. xn+1 represents the next x-value that you are trying to
find. Essentially, f'(x), the derivative represents f(x)/dx (dx = delta-x). Therefore, the term
f(x)/f'(x) represents a value of dx.