A devotee of Newton Raphson used the method to solve an equation x^100=0, using the initial estimate x_o=0. Calculate next five Newton method and Bisection estimates.
The document discusses obtaining the canonical disjunctive and conjunctive forms of a logic function. It provides an example logic table and determines the disjunctive and conjunctive forms. It then discusses Karnaugh maps as a graphical tool to simplify logic equations and minimize switching functions. It provides steps for constructing a Karnaugh map, including numbering cells and representing functions and cubes on the map. Finally, it gives an example of representing a logic function on a Karnaugh map.
This document discusses inflection points and provides two examples. An inflection point is a point on a curve where the curvature changes from increasing to decreasing or vice versa. The second derivative test can be used to find inflection points by finding where the second derivative is equal to zero. The first example finds that the function f(x)=x^3 has an inflection point at x=0, as the second derivative is zero there and changes sign on either side. The second example finds that the function f(x)=x^4 has no inflection points, as its second derivative is always positive.
Statistik 1 5 distribusi probabilitas diskritSelvin Hadi
This document discusses discrete probability distributions. It defines key terms like probability distribution, random variables, and types of random variables. It also covers calculating the mean, variance, and standard deviation of discrete probability distributions. Specific discrete probability distributions covered include the binomial, hypergeometric, and Poisson distributions. Examples are provided to demonstrate calculating probabilities and distribution properties.
The document discusses tangent planes and normal lines to surfaces. It defines a tangent plane at a point P on a surface z=f(x,y) as having an equation involving the partial derivatives of f at P. A normal line to a curve at a point P is perpendicular to the tangent line at P, with slope given by the negative reciprocal of the tangent slope. The normal line to a surface z=f(x,y,z) at a point P passes through P with direction given by the gradient of f at P.
The document discusses concepts related to calculus including tangent planes, normal lines, and linear approximations. It provides definitions and equations for calculating the tangent plane to a surface, the normal line to a curve or surface, and using the tangent line as a linear approximation near a given point on a function. Examples are given to demonstrate finding the total derivative of a function and using the tangent line as a linearization.
The document is notes for a lesson on tangent planes. It provides definitions of tangent lines and planes, formulas for finding equations of tangent lines and planes, and examples of applying these concepts. Specifically, it defines that the tangent plane to a function z=f(x,y) through the point (x0,y0,z0) has normal vector (f1(x0,y0), f2(x0,y0),-1) and equation f1(x0,y0)(x-x0) + f2(x0,y0)(y-y0) - (z-z0) = 0 or z = f(x0,y0) +
1. The document discusses the concept of the derivative and differentiation using the first principle. It explains how to calculate the slope of a tangent line to a curve at a point using limits, which gives the derivative of the function at that point.
2. Rules for differentiating common functions like polynomials, exponentials, and logarithms are covered. Higher-order derivatives and applications of derivatives to business and economics are also mentioned.
3. The goals of the class are to explain the concept of the derivative, differentiate functions using the first principle (limits), and understand various differentiation rules.
The document discusses obtaining the canonical disjunctive and conjunctive forms of a logic function. It provides an example logic table and determines the disjunctive and conjunctive forms. It then discusses Karnaugh maps as a graphical tool to simplify logic equations and minimize switching functions. It provides steps for constructing a Karnaugh map, including numbering cells and representing functions and cubes on the map. Finally, it gives an example of representing a logic function on a Karnaugh map.
This document discusses inflection points and provides two examples. An inflection point is a point on a curve where the curvature changes from increasing to decreasing or vice versa. The second derivative test can be used to find inflection points by finding where the second derivative is equal to zero. The first example finds that the function f(x)=x^3 has an inflection point at x=0, as the second derivative is zero there and changes sign on either side. The second example finds that the function f(x)=x^4 has no inflection points, as its second derivative is always positive.
Statistik 1 5 distribusi probabilitas diskritSelvin Hadi
This document discusses discrete probability distributions. It defines key terms like probability distribution, random variables, and types of random variables. It also covers calculating the mean, variance, and standard deviation of discrete probability distributions. Specific discrete probability distributions covered include the binomial, hypergeometric, and Poisson distributions. Examples are provided to demonstrate calculating probabilities and distribution properties.
The document discusses tangent planes and normal lines to surfaces. It defines a tangent plane at a point P on a surface z=f(x,y) as having an equation involving the partial derivatives of f at P. A normal line to a curve at a point P is perpendicular to the tangent line at P, with slope given by the negative reciprocal of the tangent slope. The normal line to a surface z=f(x,y,z) at a point P passes through P with direction given by the gradient of f at P.
The document discusses concepts related to calculus including tangent planes, normal lines, and linear approximations. It provides definitions and equations for calculating the tangent plane to a surface, the normal line to a curve or surface, and using the tangent line as a linear approximation near a given point on a function. Examples are given to demonstrate finding the total derivative of a function and using the tangent line as a linearization.
The document is notes for a lesson on tangent planes. It provides definitions of tangent lines and planes, formulas for finding equations of tangent lines and planes, and examples of applying these concepts. Specifically, it defines that the tangent plane to a function z=f(x,y) through the point (x0,y0,z0) has normal vector (f1(x0,y0), f2(x0,y0),-1) and equation f1(x0,y0)(x-x0) + f2(x0,y0)(y-y0) - (z-z0) = 0 or z = f(x0,y0) +
1. The document discusses the concept of the derivative and differentiation using the first principle. It explains how to calculate the slope of a tangent line to a curve at a point using limits, which gives the derivative of the function at that point.
2. Rules for differentiating common functions like polynomials, exponentials, and logarithms are covered. Higher-order derivatives and applications of derivatives to business and economics are also mentioned.
3. The goals of the class are to explain the concept of the derivative, differentiate functions using the first principle (limits), and understand various differentiation rules.
Euler's method is a numerical approach for approximating solutions to differential equations. It works by taking an initial condition and using the tangent line at that point to take a small step to a new point. This process is repeated, using the new point as the initial condition. The smaller the step size, the more accurate the approximation will be. An example walks through applying Euler's method to the differential equation y' = x + y with an initial condition of y(0) = 2 using 10 steps of size 0.1.
The document describes a C program that uses Euler's method to solve a differential equation for values of x from 0 to 0.4 in increments of 0.05. The program calculates the value of y for each x value using the Euler method formula, printing the results to 5 decimal places. It is given that the initial value of y is 0 when x is 0.
The document describes three numerical integration rules - the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule. For each rule, it provides the MATLAB code to calculate the definite integral of a function over an interval using that respective rule. The code takes the initial and final inputs, number of intervals, and function as inputs from the user. It then calculates the integral based on the specific rule and outputs the result.
The document describes MATLAB functions for numerical integration using various rules:
1) Trapezoidal, Simpsons 1/3, Simpsons 3/8, Booles, Weddles, and Rectangular rules are presented with code examples.
2) Each function takes the initial and final inputs, number of intervals, and function as inputs and returns the numerical integration as the output.
3) The functions break the interval into subintervals and calculate the area using the specific rule's formula to approximate the definite integral.
IB Maths.Turning points. Second derivative testestelav
The document discusses methods for finding maximum, minimum, and points of inflection on a curve:
1. Use the first derivative test to find stationary points where f'(a) = 0, then examine the sign of f' left and right of a to determine if it is a maximum or minimum.
2. Use the second derivative test where f'(a) = 0, and if f''(a) < 0 it is a maximum, f''(a) > 0 it is a minimum, and if f'' changes sign at a it is a point of inflection.
Several examples are provided to demonstrate finding stationary points and determining their nature using these two methods, as well as sketching
The document discusses key concepts related to functions including: if the derivative is positive the function is increasing, if negative then decreasing, critical numbers are where the derivative is 0 or doesn't exist, points of inflection are where a curve changes concavity, and an example problem is given to find the local minimum of a cubic function by taking its derivative.
Lesson3.1 The Derivative And The Tangent Lineseltzermath
This document provides an introduction to the concept of the tangent line and derivative. It defines the tangent line as the line that intersects a curve at exactly one point. It discusses how to approximate the slope of a tangent line using secant lines and taking the limit as the second point approaches the point of tangency. The derivative is defined as the formula that gives the slope of the tangent line at any point on a curve. It provides examples of using calculators to calculate derivatives and discusses how the graph of a derivative relates to properties of the original function such as maxima, minima and x-intercepts.
The document discusses using the second derivative test to determine the concavity and points of inflection of functions. It explains that the second derivative test involves finding intervals where the second derivative is positive or negative to determine if a graph is concave up or down. Points of inflection occur when the second derivative is equal to zero. Examples are provided to demonstrate how to apply this test to locate intervals of concavity and points of inflection of various functions.
This Maple code performs steepest descent optimization to find the minimum of a function. It defines the function f(x,y)=x^2-y, initializes starting values for x and y, calculates the partial derivatives at each step, finds the steepest descent direction, takes a step in that direction, and repeats for 20 iterations, printing the current values and gradient at each step.
The document outlines key calculus concepts including:
- Functions, derivatives, differentiation rules, and the definition of a derivative as an infinitesimal change in a function with respect to a variable.
- Concepts related to derivatives such as local/absolute extrema, critical points, increasing/decreasing functions, concavity, asymptotes, and inflection points.
- How to use the first and second derivative tests to determine local extrema, concavity, and increasing/decreasing behavior.
This document provides a study guide for inferential statistics. It introduces key concepts such as random variables, discrete and continuous random variables, and probability distributions. It discusses probability mass functions for discrete random variables and probability density functions for continuous random variables. It also defines important parameters of a distribution such as expected value, variance, and standard deviation. Examples are provided to illustrate how to calculate these parameters from probability distributions.
The document discusses several methods for analyzing brain data including:
1. PageRanking, which involves transforming fMRI data into a graph and investigating PageRank values to identify functional brain regions.
2. Granger causality analysis, which uses a vector autoregressive model on fMRI time series data to identify directional influences between brain areas.
3. Clustering algorithms like k-means, normalized cut, and heat kernel PageRank that are being explored to automatically identify the locations of functional brain regions from fMRI data with the goal of addressing inconsistencies between previous studies.
A polynomial function can be expressed as a sum of terms with non-negative integer exponents. A quadratic function is a polynomial of degree 2 in the form p(x)=ax^2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola, which is symmetric about the axis of symmetry and has a unique vertex. Changing the coefficients a, b, and c changes the position, shape, and orientation of the parabola.
The document discusses creating and populating database tables to store and manage employee data. It includes steps to:
1) Create tables for employees, departments, positions, and education levels using the CREATE TABLE statement.
2) Populate the employee table with sample data using the INSERT statement.
3) Grant user access to view and manage the tables.
The document discusses two interpolation methods - Newton forward difference method and backward method. It provides the code to calculate the forward difference table using Newton forward difference method. It also provides functions to implement backward interpolation method and Stirling's interpolation formula.
This document provides an example of writing the equation for a sinusoidal graph in the form y = a sin b(x + c) + d. The graph shown has an amplitude (a) of 2, angular frequency (b) of 1, phase shift (c) of 0, and vertical shift (d) of 1. Therefore, the equation for the graph is y = 2 sin x + 1.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods for finding the roots of single non-linear equations. The Newton-Raphson method uses derivatives while the secant method approximates derivatives. Both methods are demonstrated through MATLAB code and examples. The document also discusses applying the Newton-Raphson method to systems of non-linear equations by evaluating the Jacobian matrix of partial derivatives.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods for finding the roots of single non-linear equations. The Newton-Raphson method uses derivatives while the secant method approximates derivatives. Both methods are demonstrated through MATLAB code and examples. The document also discusses applying the Newton-Raphson method to systems of non-linear equations by evaluating the Jacobian matrix.
Euler's method is a numerical approach for approximating solutions to differential equations. It works by taking an initial condition and using the tangent line at that point to take a small step to a new point. This process is repeated, using the new point as the initial condition. The smaller the step size, the more accurate the approximation will be. An example walks through applying Euler's method to the differential equation y' = x + y with an initial condition of y(0) = 2 using 10 steps of size 0.1.
The document describes a C program that uses Euler's method to solve a differential equation for values of x from 0 to 0.4 in increments of 0.05. The program calculates the value of y for each x value using the Euler method formula, printing the results to 5 decimal places. It is given that the initial value of y is 0 when x is 0.
The document describes three numerical integration rules - the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule. For each rule, it provides the MATLAB code to calculate the definite integral of a function over an interval using that respective rule. The code takes the initial and final inputs, number of intervals, and function as inputs from the user. It then calculates the integral based on the specific rule and outputs the result.
The document describes MATLAB functions for numerical integration using various rules:
1) Trapezoidal, Simpsons 1/3, Simpsons 3/8, Booles, Weddles, and Rectangular rules are presented with code examples.
2) Each function takes the initial and final inputs, number of intervals, and function as inputs and returns the numerical integration as the output.
3) The functions break the interval into subintervals and calculate the area using the specific rule's formula to approximate the definite integral.
IB Maths.Turning points. Second derivative testestelav
The document discusses methods for finding maximum, minimum, and points of inflection on a curve:
1. Use the first derivative test to find stationary points where f'(a) = 0, then examine the sign of f' left and right of a to determine if it is a maximum or minimum.
2. Use the second derivative test where f'(a) = 0, and if f''(a) < 0 it is a maximum, f''(a) > 0 it is a minimum, and if f'' changes sign at a it is a point of inflection.
Several examples are provided to demonstrate finding stationary points and determining their nature using these two methods, as well as sketching
The document discusses key concepts related to functions including: if the derivative is positive the function is increasing, if negative then decreasing, critical numbers are where the derivative is 0 or doesn't exist, points of inflection are where a curve changes concavity, and an example problem is given to find the local minimum of a cubic function by taking its derivative.
Lesson3.1 The Derivative And The Tangent Lineseltzermath
This document provides an introduction to the concept of the tangent line and derivative. It defines the tangent line as the line that intersects a curve at exactly one point. It discusses how to approximate the slope of a tangent line using secant lines and taking the limit as the second point approaches the point of tangency. The derivative is defined as the formula that gives the slope of the tangent line at any point on a curve. It provides examples of using calculators to calculate derivatives and discusses how the graph of a derivative relates to properties of the original function such as maxima, minima and x-intercepts.
The document discusses using the second derivative test to determine the concavity and points of inflection of functions. It explains that the second derivative test involves finding intervals where the second derivative is positive or negative to determine if a graph is concave up or down. Points of inflection occur when the second derivative is equal to zero. Examples are provided to demonstrate how to apply this test to locate intervals of concavity and points of inflection of various functions.
This Maple code performs steepest descent optimization to find the minimum of a function. It defines the function f(x,y)=x^2-y, initializes starting values for x and y, calculates the partial derivatives at each step, finds the steepest descent direction, takes a step in that direction, and repeats for 20 iterations, printing the current values and gradient at each step.
The document outlines key calculus concepts including:
- Functions, derivatives, differentiation rules, and the definition of a derivative as an infinitesimal change in a function with respect to a variable.
- Concepts related to derivatives such as local/absolute extrema, critical points, increasing/decreasing functions, concavity, asymptotes, and inflection points.
- How to use the first and second derivative tests to determine local extrema, concavity, and increasing/decreasing behavior.
This document provides a study guide for inferential statistics. It introduces key concepts such as random variables, discrete and continuous random variables, and probability distributions. It discusses probability mass functions for discrete random variables and probability density functions for continuous random variables. It also defines important parameters of a distribution such as expected value, variance, and standard deviation. Examples are provided to illustrate how to calculate these parameters from probability distributions.
The document discusses several methods for analyzing brain data including:
1. PageRanking, which involves transforming fMRI data into a graph and investigating PageRank values to identify functional brain regions.
2. Granger causality analysis, which uses a vector autoregressive model on fMRI time series data to identify directional influences between brain areas.
3. Clustering algorithms like k-means, normalized cut, and heat kernel PageRank that are being explored to automatically identify the locations of functional brain regions from fMRI data with the goal of addressing inconsistencies between previous studies.
A polynomial function can be expressed as a sum of terms with non-negative integer exponents. A quadratic function is a polynomial of degree 2 in the form p(x)=ax^2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola, which is symmetric about the axis of symmetry and has a unique vertex. Changing the coefficients a, b, and c changes the position, shape, and orientation of the parabola.
The document discusses creating and populating database tables to store and manage employee data. It includes steps to:
1) Create tables for employees, departments, positions, and education levels using the CREATE TABLE statement.
2) Populate the employee table with sample data using the INSERT statement.
3) Grant user access to view and manage the tables.
The document discusses two interpolation methods - Newton forward difference method and backward method. It provides the code to calculate the forward difference table using Newton forward difference method. It also provides functions to implement backward interpolation method and Stirling's interpolation formula.
This document provides an example of writing the equation for a sinusoidal graph in the form y = a sin b(x + c) + d. The graph shown has an amplitude (a) of 2, angular frequency (b) of 1, phase shift (c) of 0, and vertical shift (d) of 1. Therefore, the equation for the graph is y = 2 sin x + 1.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods for finding the roots of single non-linear equations. The Newton-Raphson method uses derivatives while the secant method approximates derivatives. Both methods are demonstrated through MATLAB code and examples. The document also discusses applying the Newton-Raphson method to systems of non-linear equations by evaluating the Jacobian matrix of partial derivatives.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods for finding the roots of single non-linear equations. The Newton-Raphson method uses derivatives while the secant method approximates derivatives. Both methods are demonstrated through MATLAB code and examples. The document also discusses applying the Newton-Raphson method to systems of non-linear equations by evaluating the Jacobian matrix.
This document presents methods for solving single non-linear equations and systems of non-linear equations numerically. It describes the Newton-Raphson and secant methods. The Newton-Raphson method uses iterative approximations based on the Taylor series expansion and its derivative to find roots. The secant method approximates the derivative as the difference quotient. MATLAB functions are provided to implement both methods to find roots of sample non-linear equations.
This document discusses optimizing functions of two variables. It defines relative extrema for functions of two variables and introduces critical points, which are points where the partial derivatives are both equal to zero. These critical points can indicate relative maxima, minima, or saddle points. The document presents the Second Partials Test, which uses the values and signs of the second-order partial derivatives at a critical point to determine whether it is a relative extremum (maximum or minimum) or a saddle point. Examples demonstrate applying this test to classify critical points.
This document discusses numerical methods for finding roots of equations, specifically the secant method and Regula Falsi method. The secant method uses two initial approximations to determine the secant line of the function, and finds subsequent approximations by setting this line equal to zero. The Regula Falsi method combines bisection and secant, using the secant formula but checking signs as in bisection to refine the interval. Both methods converge faster than bisection but can fail if starting points are poor, while Regula Falsi always converges since it keeps the solution bracketed. The document also notes that these derivative-free methods are useful when the function is defined by experiments rather than a formula.
Interpolation techniques - Background and implementationQuasar Chunawala
This document discusses interpolation techniques, specifically Lagrange interpolation. It begins by introducing the problem of interpolation - given values of an unknown function f(x) at discrete points, finding a simple function that approximates f(x).
It then discusses using Taylor series polynomials for interpolation when the function value and its derivatives are known at a point. The error in interpolation approximations is also examined.
The main part discusses Lagrange interpolation - given data points (xi, f(xi)), there exists a unique interpolating polynomial Pn(x) of degree N that passes through all the points. This is proved using the non-zero Vandermonde determinant. Lagrange's interpolating polynomial is then introduced as a solution.
This document discusses linear multistep methods for solving initial value differential problems. It makes three key points:
1) Linear multistep methods can be viewed as fixed point iterative methods with a differential operator instead of an integral operator. They converge to a fixed point if consistent and stable.
2) The document proves that any linear multistep method defines a contraction mapping, ensuring convergence to a unique fixed point when the problem is Lipschitz continuous.
3) Examples of explicit and implicit linear multistep methods are given, along with their fixed point iterative formulations. These include Euler's method, trapezoidal method, and Adams-Moulton method.
The document discusses partial derivative equations and homogeneous functions. It defines:
- The partial derivative of a function f(x,y) with respect to x and y at a point.
- Clairaut's theorem, which relates mixed partial derivatives to commutative properties.
- The Laplace equation in 2D and 3D, which relates second order partial derivatives of a function.
- Conditions for a function of two variables to be homogeneous of a given degree.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
This document discusses using the Newton-Raphson iterative method to solve chemical equilibrium problems. It begins by introducing fixed point theory and the Newton-Raphson method for solving nonlinear equations. It then describes applying this method to determine the O reactant ratio that produces an adiabatic equilibrium temperature in the chemical reaction of partial methane oxidation. Specifically, it develops a system of seven nonlinear equations and uses the Newton-Raphson method to iteratively solve for the fixed point and desired chemical equilibrium conditions.
This document discusses different types of discrete probability distributions:
- The uniform distribution where all outcomes are equally likely. Rolling a fair die is given as an example.
- The Bernoulli distribution which has only two possible outcomes (success/failure) with probabilities p and q=1-p.
- The binomial distribution which models experiments with a fixed number of trials, two possible outcomes per trial (success/failure), and where trials are independent.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if |g'(α)| < 1, where α is the root and g' is the derivative of g. This ensures the error decreases at each iteration.
S3. Examples show the method can converge rapidly, as in Newton's method, or diverge, depending on the properties of g near the root. Aitken extrapolation can provide a better estimate of the root than the current iterate xn.
S1. Fixed point iteration is a numerical method for solving equations of the form x = g(x) by making an initial guess x0 and repeatedly substituting xn into the right side to obtain xn+1.
S2. The method converges if g(x) is continuous and λ, the maximum absolute value of the derivative of g(x), is less than 1.
S3. Examples show that fixed point iteration can converge slowly if the derivative of g(x) at the root is close to 1, and Aitken's method can be used to accelerate convergence by extrapolating the iterates.
This document analyzes the diophantine equation x^2 + y^2 = n^2 in connection with a famous problem proposed at the 1988 International Mathematical Olympiad. It defines a function F(x,y) and uses properties of F to find integer solutions to the equation. It shows that for prime numbers p, the only integer solutions are the trivial ones (0,p^2) and (p^3,p^2). More generally, it finds that for any number n, the only integer solutions are n^2 or possibly another non-trivial integer if n belongs to a specific sequence generated by another number. It concludes by conjecturing the full set of integer solutions.
The document discusses solutions to problems using the Newton-Raphson method for finding roots of equations. It provides solutions to 7 example problems, calculating multiple iterations of the Newton-Raphson method to approximate roots. The document also notes some limitations of calculators and computers in performing complex calculations to finite precision.
The document introduces numerical methods for finding the roots or zeros of equations of the form f(x) = 0, where f(x) is an algebraic or transcendental function. It focuses on the bisection method, also called the Bolzano method, which uses interval bisection to bracket the root between two values where f(x) has opposite signs. The method iteratively narrows down the interval to find the root to within a specified tolerance. Several examples demonstrate applying the bisection method to find roots of polynomial, logarithmic, and trigonometric equations.
The Newton-Raphson method estimates roots of equations by:
1) Rearranging the equation into the form f(x) = 0 and choosing an initial x-value
2) Substituting into the formula xn+1 = xn - f(xn)/f'(xn)
3) Differentiating to find f'(x) and iterating the formula using a calculator until convergence
The method may fail if the starting value is near a stationary point where f'(x) = 0, causing division by zero in the formula.
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End-to-end pipeline agility - Berlin Buzzwords 2024Lars Albertsson
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2. PROBLEM STATEMENT:
A devotee of Newton Raphson used the method to solve an equation x ^ 100
= 0, using the initial estimate xo=0. Calculate next fve Newton method and
Bisection estimates.
PLOT FORf (x)=x
100
:
DERIVATIVE OFf (x)=x
100
:
f
'
(x)=100 x
99
NEWTON REPHSON METHOD FORMULA:
3. xi+1=xi−
f (xi)
f
'
(xi)
, f
'
(xi )≠0
NEWTON RAPHSON METHOD IMPLIMENTATION:
Given that xo=0 is the initial guess, thereforef
'
(x)=0 , the method works only
iff ' (x)≠0.
Let’s change our assumption as xo=0.6 such thatf ' (x)≠0. The next fve iterations of
the Newton Raphson Method can be implemented as under;
% Let y = f(x) = 0 be our function such that;
y = @(x) x .^ 100;
% Let's plot the graph to analyse the behavior of y(x) for 'x'
figure;
X = -10:0.001:+10;
title('f(x)=x^{100}');
plot(X, X .^ 100);
xo = 0.1; % Initial estimate
% differentiate 'y' w.r.t. x such that;
diff_y = @(x) 100 * x .^ 99.0;
% Now, let's analyse the first five iterations
XN = [0, 0, 0, 0, 0];
YN = [0, 0, 0, 0, 0];
for iteration = 1:5
if diff_y(xo) == 0
disp('Error, Derivative is zero!')
break;
else
% let x be the next value and xo be the previous one;
% according to NRM, next value of x can be given as below
x = xo - y(xo)/diff_y(xo);
% replace the prev value with the next value for next iteration
xo = x;
% store the results to observe in near future
XN(iteration) = xo;
YN(iteration) = y(xo); % approx values of y(x)
% display the new estimate for x and respective f(x)
fprintf('x_%d = %f, y(%f) = %fn', iteration, x, x, y(x));
end
end
OUTPUT:
x_1 = 0.099000, y(0.099000) = 0.000000
x_2 = 0.098010, y(0.098010) = 0.000000
x_3 = 0.097030, y(0.097030) = 0.000000
4. x_4 = 0.096060, y(0.096060) = 0.000000
x_5 = 0.095099, y(0.095099) = 0.000000
PLOT FORf (x)=x
99
:
DERIVATIVE OFf (x)=x
99
:
f
'
(x)=99 x
98
BISECTION METHOD:
For x ∈[a,b],midpoint=
a+b
2
if f (a)f (midpoint )<0:b=midpoint ,else:a=midpoint
5. BISECTION METHOD IMPLIMENTATION:
Let x∈[−1,6], therefore a=−1 and b=6 are satisfying f (a)f (b)<0. The frst fve
iterations can be observed using the following set of commands in MATLAB;
% Let y = f(x) = 0 be our function such that;
y = @(x) x .^ 99;
% Let's plot the graph to analyse the behavior of y(x) for 'x'
figure;
X = -10:0.001:+10;
title('f(x)=x^{99}');
plot(X, X .^ 99);
% initially we assume that 'x' lies in range [a, b] where;
a = -1; b = 6; % since y(a)y(b) < 0
% Now, let's analyse the first five iterations
XB = [0, 0, 0, 0, 0];
YB = [0, 0, 0, 0, 0];
for iteration = 1:5
% mid point of interval [a,b]
x = (a+b)/2.0;
% display the new estimate for x and respective f(x)
fprintf('x_%d = %f, y(%f) = %fn', iteration, x, x, y(x));
% store the results to observe in near future
XB(iteration) = x;
YB(iteration) = y(x); % approx values of y(x)
if y(a) * y(x) < 0
b = x;
elseif y(x) * y(b) < 0
a = x;
end
end
OUTPUT:
x_1 = 2.500000, y(2.500000) =
2489206111144456600000000000000000000000.000000
x_2 = 0.750000, y(0.750000) = 0.000000
x_3 = -0.125000, y(-0.125000) = -0.000000
x_4 = 0.312500, y(0.312500) = 0.000000
x_5 = 0.093750, y(0.093750) = 0.000000
6. ANALYSIS OF THE APPROXIMATIONS:
In the above graph, ‘+’ represent the values of ‘f(x)’ at the approximated
values x in Bisection Method, while the ‘*’ represents the values of y=x
99
at the
respective values of x approximated using the Bisection Method.
7. The behavior of the Newton Raphson Method can be analyzed more clearly
by zooming the portion of graph in some range of x as under;
In the above graph we can see that for all the approximated values of ‘x’ we
are getting ‘f(x)’ approaching zero.
CONCLUSION:
Newton Raphson method, for the given function converges faster than the
Bisection Method as gives more accurate values for f(x). In other words, Newton
method can be seen more accurate than the Bisection Method, and is more
convergent as well.