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.
This document summarizes research on the consistency and stability of linear multistep methods for solving initial value differential problems. It discusses the local truncation error and consistency conditions for convergence. The consistency condition requires that the truncation error approaches zero as the step size decreases. Stability conditions like relative and weak stability are also analyzed. It is shown that linear multistep methods satisfy the conditions of the Banach fixed point theorem, ensuring a unique solution. Specifically, a two-step predictor-corrector method is presented where the predictor provides an initial estimate that is corrected.
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
The document contains solutions to 4 problems posed at the IMC 2016 conference in Bulgaria.
The first problem proves that the sum of a sequence of positive numbers divided by increasing powers of 2 is less than or equal to 2. The second problem finds the minimum value of a function over continuous functions satisfying a given inequality.
The third problem proves that if a function satisfies three properties related to permutations, then the size of the ring it is defined over must be congruent to 2 modulo 4.
The fourth problem proves an inequality relating the number of integer solutions to an inequality when the upper bound is increased or decreased by 1.
Numerical Methods and Analysis discusses various root-finding methods including bisection, false position, and Newton-Raphson. Bisection uses interval halving to find a root between two values with opposite signs. False position uses the slope of a line between two points to estimate the next root. Newton-Raphson approximates the root using Taylor series expansion neglecting higher order terms. Interpolation uses forward difference tables to construct a polynomial approximation of a function.
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices used in the iteration formula.
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices derived from the system's coefficients.
2.hyperbolic functions Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
This chapter introduces hyperbolic functions including definitions, properties, graphs and calculus applications. The hyperbolic cosine and sine functions are defined analogously to trigonometric cosine and sine using exponential functions. Hyperbolic functions have similar properties and relationships as trigonometric functions which can be represented using Osborn's rule. Their graphs have distinct shapes important for applications in calculus, including integration and solving equations. Inverse hyperbolic functions are also introduced along with their relationship to corresponding hyperbolic functions.
This document summarizes research on the consistency and stability of linear multistep methods for solving initial value differential problems. It discusses the local truncation error and consistency conditions for convergence. The consistency condition requires that the truncation error approaches zero as the step size decreases. Stability conditions like relative and weak stability are also analyzed. It is shown that linear multistep methods satisfy the conditions of the Banach fixed point theorem, ensuring a unique solution. Specifically, a two-step predictor-corrector method is presented where the predictor provides an initial estimate that is corrected.
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
The document contains solutions to 4 problems posed at the IMC 2016 conference in Bulgaria.
The first problem proves that the sum of a sequence of positive numbers divided by increasing powers of 2 is less than or equal to 2. The second problem finds the minimum value of a function over continuous functions satisfying a given inequality.
The third problem proves that if a function satisfies three properties related to permutations, then the size of the ring it is defined over must be congruent to 2 modulo 4.
The fourth problem proves an inequality relating the number of integer solutions to an inequality when the upper bound is increased or decreased by 1.
Numerical Methods and Analysis discusses various root-finding methods including bisection, false position, and Newton-Raphson. Bisection uses interval halving to find a root between two values with opposite signs. False position uses the slope of a line between two points to estimate the next root. Newton-Raphson approximates the root using Taylor series expansion neglecting higher order terms. Interpolation uses forward difference tables to construct a polynomial approximation of a function.
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices used in the iteration formula.
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices derived from the system's coefficients.
2.hyperbolic functions Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
This chapter introduces hyperbolic functions including definitions, properties, graphs and calculus applications. The hyperbolic cosine and sine functions are defined analogously to trigonometric cosine and sine using exponential functions. Hyperbolic functions have similar properties and relationships as trigonometric functions which can be represented using Osborn's rule. Their graphs have distinct shapes important for applications in calculus, including integration and solving equations. Inverse hyperbolic functions are also introduced along with their relationship to corresponding hyperbolic functions.
4 pages from matlab an introduction with app.-2Malika khalil
This document discusses several methods for solving systems of linear equations and algebraic eigenvalue problems, including Gauss elimination, LU decomposition, Cholesky decomposition, Gauss-Seidel, Gauss-Jordan, and Jacobi methods. It provides the mathematical formulations and iterative procedures for each method. Numerical examples using MATLAB are provided to illustrate how to apply the procedures.
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 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.
This document provides information about Calculus 2, including lessons on indeterminate forms, Rolle's theorem, the mean value theorem, and differentiation of transcendental functions. It defines Rolle's theorem and the mean value theorem, provides examples of applying each, and discusses how Rolle's theorem can be used to find the value of c. It also defines inverse trigonometric functions and their derivatives. The document is for MATH 09 Calculus 2 and includes exercises for students to practice applying the theorems.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
Iterative solution methods start with an initial approximation for the solution vector and iteratively update the vector at each step by using the system Ax=b. Different iterative methods like Jacobi, Gauss-Seidel, and Successive Over Relaxation represent the system in the form x=Ex+f and generate the next approximation. The iterative method converges if the spectral radius of E is less than 1 as the number of iterations increases. Different convergence criteria like the norm of the residual vector can be used to check when the iterations have converged to the solution.
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
I am Manuela B. I am a Differential Equations Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick. I have been helping students with their assignments for the past 13 years. I solve assignments related to Differential Equations.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Differential Equations Assignment.
I am Manuela B. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick Profession. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
I am Piers L. I am a Calculus Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From, the University of Adelaide. I have been helping students with their homework for the past 7 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
Solution to schrodinger equation with dirac comb potential slides
This document summarizes solving the Schrödinger equation for a Dirac comb potential. The potential is an infinite series of Dirac delta functions spaced periodically. Floquet theory is used to solve the time-independent Schrödinger equation for this potential. Boundary conditions are applied and the resulting equations are solved graphically. Allowed energy bands are determined and plotted versus wave vector for both attractive and repulsive delta function potentials.
I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Differential Equations Homework.
This document discusses three methods for finding the roots of nonlinear equations:
1) Bisection method, which converges linearly but is guaranteed to find a root.
2) Newton's method, which converges quadratically (much faster) but may diverge if the starting point is too far from the root.
3) Secant method, which is faster than bisection but slower than Newton's, and also requires starting points close to the root. Newton's and secant methods can be extended to systems of nonlinear equations.
This document describes the process of using natural and clamped cubic splines to approximate functions based on data points. It presents the mathematical formulas for natural and clamped cubic splines and their derivatives. Code functions are provided to calculate the splines and plot the results. The document demonstrates applying this process to example functions and data, showing the natural and clamped cubic splines accurately fit the original functions.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
4 pages from matlab an introduction with app.-2Malika khalil
This document discusses several methods for solving systems of linear equations and algebraic eigenvalue problems, including Gauss elimination, LU decomposition, Cholesky decomposition, Gauss-Seidel, Gauss-Jordan, and Jacobi methods. It provides the mathematical formulations and iterative procedures for each method. Numerical examples using MATLAB are provided to illustrate how to apply the procedures.
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 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.
This document provides information about Calculus 2, including lessons on indeterminate forms, Rolle's theorem, the mean value theorem, and differentiation of transcendental functions. It defines Rolle's theorem and the mean value theorem, provides examples of applying each, and discusses how Rolle's theorem can be used to find the value of c. It also defines inverse trigonometric functions and their derivatives. The document is for MATH 09 Calculus 2 and includes exercises for students to practice applying the theorems.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
Iterative solution methods start with an initial approximation for the solution vector and iteratively update the vector at each step by using the system Ax=b. Different iterative methods like Jacobi, Gauss-Seidel, and Successive Over Relaxation represent the system in the form x=Ex+f and generate the next approximation. The iterative method converges if the spectral radius of E is less than 1 as the number of iterations increases. Different convergence criteria like the norm of the residual vector can be used to check when the iterations have converged to the solution.
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
I am Manuela B. I am a Differential Equations Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick. I have been helping students with their assignments for the past 13 years. I solve assignments related to Differential Equations.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Differential Equations Assignment.
I am Manuela B. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick Profession. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
I am Piers L. I am a Calculus Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From, the University of Adelaide. I have been helping students with their homework for the past 7 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
Solution to schrodinger equation with dirac comb potential slides
This document summarizes solving the Schrödinger equation for a Dirac comb potential. The potential is an infinite series of Dirac delta functions spaced periodically. Floquet theory is used to solve the time-independent Schrödinger equation for this potential. Boundary conditions are applied and the resulting equations are solved graphically. Allowed energy bands are determined and plotted versus wave vector for both attractive and repulsive delta function potentials.
I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Differential Equations Homework.
This document discusses three methods for finding the roots of nonlinear equations:
1) Bisection method, which converges linearly but is guaranteed to find a root.
2) Newton's method, which converges quadratically (much faster) but may diverge if the starting point is too far from the root.
3) Secant method, which is faster than bisection but slower than Newton's, and also requires starting points close to the root. Newton's and secant methods can be extended to systems of nonlinear equations.
This document describes the process of using natural and clamped cubic splines to approximate functions based on data points. It presents the mathematical formulas for natural and clamped cubic splines and their derivatives. Code functions are provided to calculate the splines and plot the results. The document demonstrates applying this process to example functions and data, showing the natural and clamped cubic splines accurately fit the original functions.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
ALPHA LOGARITHM TRANSFORMED SEMI LOGISTIC DISTRIBUTION USING MAXIMUM LIKELIH...BRNSS Publication Hub
The document discusses the alpha logarithm transformed semi-logistic distribution and its maximum likelihood estimation method. It introduces the distribution, provides its probability density function and cumulative distribution function. It then describes generating random numbers from the distribution and outlines the maximum likelihood estimation method to estimate the distribution's unknown parameters. This involves deriving the likelihood function and taking its partial derivatives to obtain equations that are set to zero and solved to find maximum likelihood estimates of the location, scale, and shape parameters.
AN ASSESSMENT ON THE SPLIT AND NON-SPLIT DOMINATION NUMBER OF TENEMENT GRAPHSBRNSS Publication Hub
This document summarizes research on the split and non-split domination numbers of tenement graphs. It defines tenement graphs and provides basic definitions of domination, split domination, and non-split domination. Formulas for the split and non-split domination numbers of tenement graphs are presented based on the number of vertices. Theorems are presented stating that the mid vertex set of a tenement graph is always a split dominating set, but its size is not always equal to the split domination number.
This document summarizes research on generalized Cantor sets and functions where the standard construction is modified. It introduces Cantor sets defined by an arbitrary base where the intervals removed at each stage are not all the same length. It also defines irregular or transcendental Cantor sets generated by transcendental numbers like e. The key findings are:
1) There exists a unique probability measure for generalized Cantor sets that generates the cumulative distribution function.
2) The Holder exponent of generalized Cantor sets is shown to be logn/s where n is the base and s is the number of subintervals.
3) Lower and upper densities are defined for the measure on generalized Cantor functions and their properties are
SYMMETRIC BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE ALGEBRABRNSS Publication Hub
1) The document discusses symmetric bilinear pairings on elliptic curves and Lie algebras in the context of cryptography. It provides an overview of the theoretical foundations and applications of combining these areas.
2) Key concepts covered include the Weil pairing as a symmetric bilinear pairing on elliptic curves, its properties of bilinearity and non-degeneracy, and efficient computation. Applications of elliptic curves in cryptography like ECDH and ECDSA are also summarized.
3) The security of protocols like ECDH and ECDSA relies on the assumed difficulty of solving the elliptic curve discrete logarithm problem (ECDLP). The document proves various mathematical aspects behind symmetric bilinear pairings and their use in elliptic curve cryptography.
SUITABILITY OF COINTEGRATION TESTS ON DATA STRUCTURE OF DIFFERENT ORDERSBRNSS Publication Hub
This document summarizes research investigating the suitability of cointegration tests on time series data of different orders. The researchers used simulated time series data from normal and gamma distributions at sample sizes of 30, 60, and 90. Three cointegration tests (Engle-Granger, Johansen, and Phillips-Ouliaris) were applied to the data. The tests were assessed based on type 1 error rates and power to determine which test was most robust for different distributions and sample sizes. The results indicated the Phillips-Ouliaris test was generally the most effective at determining cointegration across different sample sizes and distributions.
Artificial Intelligence: A Manifested Leap in Psychiatric RehabilitationBRNSS Publication Hub
Artificial intelligence shows promise in improving psychiatric rehabilitation in 3 key ways:
1) AI can help diagnose and treat mental health issues through virtual therapists and chatbots, improving access and reducing stigma.
2) Technologies like machine learning and big data allow personalized interventions and more accurate diagnoses.
3) The COVID-19 pandemic has increased need for mental health support, and AI may help address gaps by providing remote services.
A Review on Polyherbal Formulations and Herbal Medicine for Management of Ul...BRNSS Publication Hub
This document provides a review of polyherbal formulations and herbal medicines for treating peptic ulcers. It discusses how peptic ulcers occur due to an imbalance between aggressive and protective factors in the gastrointestinal tract. Common causes include H. pylori infection and NSAID use. While synthetic medications are available, herbal supplements are more affordable and have fewer side effects. The review examines various herbs that have traditionally been used to treat ulcers, including their active chemical constituents. It defines polyherbal formulations as combinations of two or more herbs, which can enhance therapeutic effects while reducing toxicity. The document aims to summarize recent research on herb and polyherbal formulation treatments for peptic ulcers.
Current Trends in Treatments and Targets of Neglected Tropical DiseaseBRNSS Publication Hub
This document summarizes current trends in treatments and targets of neglected tropical diseases. It begins by stating that neglected tropical diseases affect over 1.7 billion people globally each year and are caused by a variety of microbes. The World Health Organization is working to eliminate 30 neglected tropical diseases by 2030. The document then discusses several specific neglected tropical diseases in more detail, including human African trypanosomiasis, Chagas disease, leishmaniasis, soil-transmitted helminths, and schistosomiasis. It describes the causative agents, transmission methods, symptoms, affected populations, and current treatment options for each of these diseases. Overall, the document aims to briefly discuss neglected infectious diseases and treatment
Evaluation of Cordia Dichotoma gum as A Potent Excipient for the Formulation ...BRNSS Publication Hub
This document summarizes a study that evaluated Cordia dichotoma gum as an excipient for oral thin film drug delivery. Films were prepared with varying ratios of the gum, plasticizers (methyl paraben and glycerine), and the model drug diclofenac sodium. The films were evaluated for properties like thickness, folding endurance, tensile strength, water uptake, and drug release kinetics. The results found that a film with 10% gum, 0.2% methyl paraben and 2.5% glycerine (CDF3) exhibited the best results among the formulations tested. Stability studies showed the films were stable for 30 days at different temperatures. Overall, the study demonstrated that C.
Assessment of Medication Adherence Pattern for Patients with Chronic Diseases...BRNSS Publication Hub
This study assessed medication adherence and knowledge among rural patients with chronic diseases in South Indian hospitals. 1500 hypertensive patients were divided into intervention and control groups. The intervention group received education from pharmacists at various times, while the control group did not. A questionnaire evaluated patients' medication knowledge at baseline and several follow-ups. The intervention group showed improved medication knowledge scores after education compared to the control group. Female gender, lower education, and income were linked to lower knowledge. The study highlights the need to educate rural patients to improve medication understanding and adherence.
This document proposes a system to hide information using four algorithms for image steganography. The system first encrypts data using a modified AES algorithm. It then encrypts the encrypted data using a modified RSA algorithm. Next, it uses a fuzzy stream algorithm to add ambiguity. Finally, it hides the encrypted data in the least significant bits of cover images using LSB steganography. The document evaluates the proposed system using metrics like PSNR, MSE, and SSIM to analyze image quality and the ability to hide data imperceptibly compared to other techniques. It selects four color images as cover files and tests the system on them.
The document discusses Goldbach's problems and their solutions. It summarizes that the ternary Goldbach problem, which states that every odd number greater than 7 can be represented as the sum of three odd primes, was solved in 2013. It also discusses Ramare's 1995 proof that any even number can be represented as the sum of no more than 6 primes. The document then provides proofs for theorems related to representing numbers as sums of primes and concludes there are an infinite number of twin primes.
The document summarizes research on k-super contra harmonic mean labeling of graphs. It defines k-super Lehmer-3 mean labeling of a graph as an injective vertex labeling such that the induced edge labels satisfy certain properties. It proves that several families of graphs admit k-super Lehmer-3 mean labeling for any positive integer k, including triangular snakes, double triangular snakes, alternative triangular snakes, quadrilateral snakes, and alternative quadrilateral snakes. The document introduces the concept of k-super Lehmer-3 mean labeling and investigates this property for these families of graphs.
The document summarizes research on using various iterative schemes to solve fixed-point problems and inequalities involving self-mappings and contractions in Banach spaces. It defines concepts like non-expansive mappings, mean non-expansive mappings, and rates of convergence. The paper presents two theorems: 1) an iterative scheme for a sequence involving a self-mapping T is shown to converge to a fixed point of T, and 2) an iterative process involving a self-contraction mapping T is defined and shown to converge. Limiting cases are considered to prove convergence as the number of iterations approaches infinity.
This document summarizes research on analyzing and simulating the accuracy and stability of closed-loop control systems. It discusses various techniques for evaluating accuracy and stability, including steady-state error analysis, stability analysis, and simulation. Factors that can affect accuracy and stability are also identified, such as sensor noise, model inaccuracies, and environmental disturbances. The paper provides an overview of closed-loop control systems and their uses in various engineering fields like manufacturing, chemical processes, vehicles, aircraft, and power systems.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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AJMS_389_22.pdf
1. www.ajms.com 7
ISSN 2581-3463
RESEARCH ARTICLE
On the Review of Linear Multistep Methods as Fixed Point Iterative Methods in
the Solution of a Coupled System of Initial Value Differential Problem
Eziokwu C. Emmanuel
Departments of Mathematics, Micheal Okpara University of Agriculture, Umudike, Abia, Nigeria
Received: 27-02-2021; Revised: 20-03-2022; Accepted: 20-04-2022
ABSTRACT
The linear multi-step method, x x h f k
n k
j
k
j n j
j
k
j j n j
+
=
−
+
=
+
= + ≥
∑ ∑
0
1
0
1
α β β ; , a numerical method for solving the
initial value problem x f t x x t x
'
= ( ) ( )=
, , 0 0 is hereby analytically reviewed and observations show that
the linear multistep method is a typical Picard’s fixed point iterative formula with a differential operator
endowed with some numerical reformulations instead of the usual integral operator as in the traditional
Picard’s method. The study also shows that any given linear multistep iterative method will be convergent
to a fixed point if and only if it is consistent and stable. This was adequately illustrated with an example
as is contained in section three of this work.
Key words: Linear multistep, Fixed point iterative method, Consistence, Stability convergence
2020 Mathematics Subject Classifications: 47J26 and 46N40
Address for correspondence:
Eziokwu C. Emmanuel
E-mail: okereemm@yahoo.com
INTRODUCTION
Consider the differential equation
x f t x x t x
= ( ) ( )= …
, ; 0 0 (1.1)
A computational method for determining the
sequence{xn
} that takes the form of a linear
relationship between xn+j
, fn+j
;=0,1,…k is called
the linear multistep method of step number k or
k– step method;
f f t x f t x t
n j n j n j n j n j
+ + + + +
= ( )= ( )
( )
, , .
The general linear multistep method[1-4]
may be
given thus –
x x h f
n k
j
k
j n j
j
k
j j n j
+
=
−
+
=
+
= + …
∑ ∑
0
1
0
α β β (1.2)
Where αj
and βj
are constants, αk
≠0 and not
both α0
and β0
are zeros. We may without loss of
generality, and for the avoidance of arbitrariness
set αk
=1 throughout.
Suppose in (1.2), βk
=0 then it is called an
explicit method since it yields the current value
xn+k
directly in terms ofxn+j
, fn+j
; j = 0,1,…,k–1
which by the stage of computation have already
been calculated.[5-8]
If, however, Bk
≠0, then (1.2)
is called an implicit method which requires the
solution at each stage of the computation of the
equation
x h f t x g
n k k n k n k
+ + +
= + + …
β ( , (1.3)
Were ɡ is a known function of the previously
calculated values, xn+j
, fn+j
; j=0,1,…,k–1
If the lipschittz condition is applied onfand
m=lh|Bk
|, we observe that the unique solution
exists as seen in the theorem (3.1) and that the
unique solution for xn+k
exists as well where the
computational iteration converges to xn+k
if
h B i e h
h ²
k
k
. .
1
We, therefore, see that;[9-12]
implicit methods call
for a substantially greater deal of computational
2. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 8
efforts than the explicit method whereas, on the
other hands, for a given step number, implicit
methods can be made more accurate than the
explicit ones. Moreover, they have favorable
stability properties as will be seen in Section 3.
MAIN RESULT ON LINEAR MULTI-
STEPITERATIVE METHODS:
Analytical study of the linear multistep methods
has revealed the following facts:-
a. That the domain of existence of solution of
the linear multistep methods is the complete
metric space.
b. That the solution of the linear multistep method
converges in the complete metric space.
c. That the initial value problem x’=f(t,x);x(t0
)=x0
solvable by the linear multistep in the complete
metric space is a continuous function.
d. That the linear multistep method satisfies the
conditions of the Banach contraction mapping
principle.
e. That the linear multistep method is exactly the
Picard’s iterative method with a differential
operator instead of the usual integral operator.
Theorem 2.1: Let X be a complete metric space
and let R be a region in (t,x) plane containing(t0
,x0
)
for x0
,x ∈ X. Suppose, given
x f t x x t x
= ( ) ( )= …
, ; 0 0 (2.0)
A differential equation where f(t,x) is continuous.
If the map f in (2.1) is Lipschitzian and with
constant K1, then the initial value problem (2.1)
by the linear multistep method has a unique fixed
point.
x x x h f
n k
j
k
j n j
j
k
j n j
*
= = + …
+
=
−
+
=
+
∑ ∑
0
1
0
α β
With a two-step predictor-corrector method.
i e Thepredictor x x h f
j
j
j
n
j j
j
n
j j
. . : ( )
+
= =
= +
∑ ∑
1
0 0
α β
Thecorrector x x h f
j
j
j
n
j j
j
j
n
j j
: ( ) ( )
+
=
+
=
+ +
= +
∑ ∑
1
0
1
0
1 1
α β
For xn+k
satisfying. h
h ²
m h ²
k
k
=
1
,
Proof
Let x1
= f(x0
)
x f x f f x f x
2 1 0
2
0
= ( )= ( )
( )= ( )
x f x f f x f x
3 2
2
0
3
0
= ( )= ( )
( )= ( )
x f x f f x f x
n n
n n
= ( )= ( )
( )= ( )
−
−
1
1
0 0
x f x f f x f x
n k n k
n k n k
+ + −
+ − +
= ( )= ( )
( )= ( )…
1
1
0 0
(2.1)
We have constructed a sequence {xn
}n=0
in (X,ρ).
We shall prove that this sequence is Cauchy.
First, we compute
ρ ρ
x x f x f x
n k n k n k n k
+ + + + + +
( )=
, ( ( ), ( )
1 1 Using (2.1)
≤ Kρ(xn+k-2
,xn+k-1
) Since is a contraction
=Kρ(fn+k-2
,fn+k-1
) Using (2.1)
≤K[Kρ(xn+k-2
,xn+k-1
)] Since is a contraction
=K2 ρ
(xn+k-2
,xn+k-1
)
Kn+k
ρ(x0
,x1
)
i e K x x K x x
n k n k
n k
. . , ,
ρ ρ
+ + +
+
( )≤ ( )…
1 0 1 (2.2)
We can now show that{xn+k
}n=0
is Cauchy.
Let m+k n + k. Then
ρ ρ ρ
x x x x x x
x x
n k m k n k m k n k m k
n k m k
+ + + + + − + −
+ − +
( )≤ ( )+ ( )
+…+ ( )
, , ,
,
1 2
1
≤ ( )+ ( )
+…+ ( )
+ + −
+ −
K x x K x x
K x x
n k n k
n k
ρ ρ
ρ
0 1
1
0 1
1
0 1
, ,
, Using (2.2)
= ( ) + + +…+ + +…
+ + − +
K x x k k k k
n k n m n k
ρ 0 1
2 1
1
, ( )
Sincetheseriesontherighthandsideisageometric
progression with common ratio 1, it sum to
infinity is
1
1− k
. Hene, we have from above that
ρ ρ
x x k x x
k
asn k cek
n m
n k
, ,
sin
( )≤ ( )
−
→
− → ∞
−
0 1
1
1
0
1
Hence, the sequence {xn+k
}n=0
is a Cauchy sequence
in X and since X is complete,
{ }
xn k n
+ =
∞
0 Converges to a point in X.]
Let xn+k
→x* as h→∞… (2.3)
Since f is a contraction and hence is continuous it
follows from (2.3) that
3. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 9
f(xn+k
)→f(x*) as n→∞. But f(xn+k
)=xn+k+1
from
(2.2). So
xn+k+1
=f(xn+k
)=f(x*)… (2.4)
However, limits are unique in a metric space, so
from (2.3) and (2.4), we obtain that
f(x*)=x*… (2.5)
Hence, f has a unique fixed point in X. We shall
now prove that this fixed point is unique. Suppose
for contradiction, there exists y*∈X such that
y*=x* and (y*)=y*… (2.6)
Then, from (2.5) and (2.6)
ρ ρ ρ
x y f x f y k x y
* * * * * *
, , ( , )
( )= ( ) ( )
( )≤
So that
(k – 1) ρ(x*,y*) ≥ 0 and ρ(x*,y*)= 0.
We can divide by it to get k–1 ≥ 0, that is, k ≥ 1
which is a contradiction.
Hence x* = y* and the fixed point is unique.
Therefore,
x x h f a t n
n k
j
k
j n j
j
k
j n j n
+
=
−
+
=
+
= + ≤ ≤
∑ ∑
0
1
0
α β ;
Is the linear multistep fixed point iterative formular
for the initial value problem
x f t x ;x t x
= ( ) ( )=
, 0 0
Of the ordinary differential type.
Finally, to be sufficiently sure, we also show that
x x x h f
n k
j
k
j n j
j
k
j n j
*
= = +
+
=
−
+
=
+
∑ ∑
0
1
0
α β
Satisfies the lipschitz condition.
x y x y
* *
n k n k
− = −
+ +
= +
− +
=
−
+
=
+
=
−
+
=
∑ ∑ ∑ ∑
j
k
j n j
j
k
j n j
j
k
j n j
j
k
j n
x h f y h f
0
1
0 0
1
0
α β α β +
+
j
*
≤ − + −
=
−
+ +
=
+ +
∑ ∑
j
k
j n j n j
j
k
j n j n j
x y h f f
0
1
0
α β *
= − + −
=
−
+ +
=
+ +
∑ ∑
k x y k f f
j
k
n j n j
j
k
n j n j
*
1
0
1
2
0
= +
( ) +
=
−
+
=
+
∑ ∑
k k z f
j
k
n j
j
k
m k
1 2
0
1
0
= +
=
+ +
∑
K z f
j
k
n j m j
0
Hence, x*=xn+k
is Lipschitizian and is a continuous
map with the above fixed point.
Also in the same pattern, iterative methods for the
respective linear multistep methods are as follows:
A. The Explicit Methods are:-
i. Euler:
xn+1
=xn
+hfn
ii. The midpoints method:
xn+2
=xn
+2hfn+1
iii. Milne’s method:
x x
h
f f f
n n n n n
+ − − −
= + + +
[ ]
1 3 2 1
4
3
2 2
iv. Adam’s method:
x x
h
f f f f
n n n n n n
+ − − −
= + − + −
[ ]
1 1 2 3
24
55 59 35 9
v. The Generalized predictor method:
x x h f
j
j
j
n
j j
j
n
j j
+
= =
= +
∑ ∑
1
0 1
( )
α β
B. The Implicit Methods are:-
i. Trapezoidal method:
x x
h
f f
n
j
n n n
j
+
+
+
= + +
1
1
2
1
6
( ) ( )
ii. Simpson’s method:
x x
h
f f f
n
j
n n
j
n n
+
+
+ +
= + + +
2
1
1 1
3
4
( ) ( )
iii. Simpson’s method:
x x
h
f f f
n
j
n n n n
j
+
+
− − +
= + + +
2
1
2 1 1
3
4
( ) ( )
iv. Adams Moulton’s method
x x
h
f f f f
n
j
n n n n n
+
+
+ − −
= + + − +
[ ]
2
1
1 1 2
3
9 19 5
( )
v. Milne’s corrector method:
x x
h
f f f n
n
j
n n n n
j
+
+
− − +
= + + +
=
1
1
2 1 1
3
4 1
( ) ( )
;
vi. The Generalized corrector method
x x h f
j
j
j
n
j j
j
j
n
j j
+
=
+
=
+ +
= +
∑ ∑
1
0
1
1
1 1
( ) ( )
α β
C: The Generalized Iterative (Corrector –
Predictor) Methods are
x x h
j
j
j
n
j j
j
n
j
+
= =
= + …
∑ ∑
1
0 1
( )
α β (C1)
4. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 10
x x h f
j
j
j
n
j j
j
j
n
j j
+
=
+
=
+ +
= + …
∑ ∑
1
0
1
1
1 1
( ) ( )
α β (C2)
Here, x X
j
j
+
+
∈
1
1
( )
is the corrector points to be
determined for all j ≥ 0 while x X
j
j
+
+
∈
1
1
( )
is
predetermined before x X
j
j
+
+
∈
1
1
( )
. While the
iterations are alternatively implement one after the
other starting first with the predictor.
Note: The generalized compact form of C1 and C2
is as follows
x x h f
n k
j
k
j n j
j
k
j n j
+ −
=
−
+
=
+
= +
∑ ∑
1
0
1
0
α β
Convergence Analysis
Given the linear multistep method
x x h F a t n
n k
j
k
j n j
j
k
j n j n
+
=
−
+
=
+
= + ≤ ≤ …
∑ ∑
0
1
0
α β ; (2.7)
Theorem 2.2:[13-16]
Let xn+j
=x(tn+j
); j = 0,1,2,…,k–1 denote its
numerical solution
T x t h f t x
n k
j
k
j n j
j
k
j n j n j
+
=
−
+
=
+ +
= ( )− ( )
∑ ∑
0
1
0
α β ( , )
The local truncation error and
τn k n k
h
T x
+ +
=
1
( )
Then, the linear multistep method (3.1) is said to
be consistent if
τ(h)=max|Tn+k
(x)|→0 as h→0 and ∑ (h) =0(hm
)
For some m ≥ 1 or equivalently (1.3.2) is said to
be consistent if
j
k
j
j
k
j
j
j
j
= = =
∑ ∑ ∑
+ = …
0 0 1
1
α α β
; (2.8)
Proof:
If the numerical solution of a given linear multistep
method is
x x t j k
n j n j
+ +
= = … − …
( ), , , , ,
0 1 2 1 (2.9)
Moreover, the local truncation error is
T x t h f t x
n k
j
k
j n j
j
k
j n j n j
+
=
−
+
=
+ +
= ( )− ( )…
∑ ∑
0
1
0
α β , (2.10)
With τn k n k
x
h
T x
+ +
( ) = ( ) …
1
, (2.11)
We want to prove that the linear multistep method
x x h f a t n
n k
j
k
j n j
j
k
j n j n j
+
=
−
+
=
+ +
= + ≤ ≤
∑ ∑
0
1
0
α β , (2.12)
Is consistent if
τ h T x ash
n k
( ) = → → …
+
max ( ) 0 0 (2.13)
And
τ(h)= 0(hm
) For some m ≥ 1… (2.14)
If x ̅n+j
denotes the numerical solution with the
above exact values (1.2), then (2.14) yields
x x h f t x h f t x
n k
j
k
j n j k n k n k
k
k
n j n j
+
=
−
+ + +
=
−
+ +
+ = ( )+ ( )…
∑ ∑
0
1
0
1
α β β
, ,
(2.15)
Applying localizing assumption on (2.15) means
that no previous truncation error has been made
and that
x x t j k
n j n j
+ +
= ( ) = … −
, , , ,
0 1 1
So that we have
x x t h f t x
h f t x
n k
j
k
j n j k n k n k
k
k
j n j n j
+
=
+ + +
=
−
+ +
+ = ( )
+ ( )
∑
∑
0
0
1
α β
β
( ) ,
, …
… (2.16)
Using the local truncation error earlier defined in
(2.2), we now have
x t x t T h f t x t
h
n k
j
k
j n j n k k n k n k
k
k
( ) ( )) )), (
+
=
−
+ + + +
=
−
+ = + ( )
+
∑
0
1
0
1
α β
∑
∑ + +
( )
βj n j n j
f t x t
, ( ) ...
(2.17)
Subtracting (2.16) from (2.17) we have
x t x T h f t x t
f t x
n k n k n k k n k n k
n k n k
( ) [ , ( )
, ...
+ + + + +
+ +
− = + ( )
− ( )
β
(2.18)
If we apply mean value theorem on (3.1.10),
We have
f t x t f t x
x t x
f
x
t
n k n k n k n k
n k n k n k n k
+ + + +
+ + + +
( )− ( )=
( )−
( )∂
∂
, ( ) ,
( )
,τ
Where ηn+k
lies between x ̅n+k
and x(tn+k
)
Therefore,
5. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 11
1−
∂
∂
( )
( )
( )= …
+ + + + +
h
f
x
t x t x T
k n k n k n k n k n k
β η
, ,
(2.19)
Let en+k
represent the error at (n+k) point, so that if
the method is explicit βk
=0 and then Tn+k
=en+k
but
if the method is implicit βk
≠0 and
h
f
x
t n
k n k n k
β
∂
∂
( )
+ +
, is small then Tn+k
≈ en+k
Again let
τ n k n k
x
h
T x
+
( ) +
( ) = ( )…
1
(2.20)
For us to show that the approximate solution
{xn
|t0
≤ tn
≤ b}┤of (3.1.4) converges to the
theoretical solution x(t) of the initial value problem
x f t x x t x
= ( ) ( )=
, ; 0 0
We[13,14]
need to necessarily satisfy the consistency
condition
τ h T x ash
t t b
n k
n
( ) = → → …
≤ ≤
+
max ( )
0
0 0 (2.21)
Plus the condition that
τ(h) = 0(hm
), for some m ≥ 1… (2.22)
By this, we[15]
show the only necessary and
sufficient condition for the linear multistep (2.14)
to be consistent is that
j
k
j
j
k
j
j
k
j
and j
= = =
∑ ∑ ∑
= − + = …
0 0 1
1 1
α α β (2.23)
And for (2.23) above to be valid for all functions
x(t) is for x(t) that are m + 1 times continuously
differentiable to necessarily satisfy
j
k
j
j
k
i
j
j j i m
= =
−
∑ ∑
− + − = = … …
0 1
1
1 2
( ) ( ) , ,
α β (2.24)
Hence, we know that
T x w T x T w
n k n k n k
+ + +
+
( ) = ( )+ ( )…
α β α β (2.25)
For all constants a,b and all differentiable
functions x,w. we now examine the consequence
of (2.19) and (2.20) by expanding x(t) about tn
using Taylor’s theorem and we have
x t
j
t t x t R t
j o
k
n m
( ) = −
( ) ( )+ ( )
=
+
∑
1
0
1
1 (2.26)
Assuming x(t) is m+1 times continuously
differentiable. Substituting into the truncation
error
T x x t x t h F t
n k n k
j
k
j n j
j
k
j n j
+ +
=
+
=
+
( ) = ( )− ( )+ ( )…
∑ ∑
0 1
α β
(2.27)
And also using (2.23)
T x
j
x t T t t T R
n k
j
m
j
n n k n
j
n k m
+
=
−
+ + +
( ) = ( ) −
( )+ …
∑
0
1
1
1 ( )
( ) ( )
(2.28)
It becomes necessary[16,17]
; Keller[18]
to calculate
Tn+k
(t–tn
)j
For j=0
T c
n k
j
k
j
+
=
( ) = ≡ − …
∑
1 1
0
0
α (2.29)
For j ≥ 1 we have
T t t T t t
n k n
j
n k n
j
+ +
− = −
( ) ( )
= − + −
= …
=
+
=
+
=
∑ ∑
j
k
j n k n
j
j
k
j
i
n j n
i
t t h t t c h
0 0
1
1
1
α β
( ) ( )
(2.30)
C j i j i
j
j
k
i
j
j
k
i
j
= − − + −
≥
= =
=
∑ ∑
1 1
0 1
1
( ) ( ) ,
α β
This gives
T x
c
j
h x t T R
n k
j
m
j j
n n k m
+
=
+ +
( ) = ( )+ ( )…
∑
1
1
( )
(2.31)
Moreover, if we write the remainder Rm+1
(t) as
R t
m
t t x t
m n
m m
n
+
+ +
( ) =
+
( )
− ( )+…
1
1 1
1
1 !
( )
Then,
T R t
C
m
h x t h
n k m
m m m
n
m
+ +
+ + + +
( ) =
+
( )
( )+ …
1
1 1 1 2
1
0
!
( )
(2.32)
To obtain the consistency condition (2.20), we
need τ(h)– 0(h) and this requires Tn+k
(x)=0(h2
).
Using (2.19) with m=1, we must have C0
,C1
=0
which gives the set of equations (2.22) which are
referred to as consistency conditions in some texts.
Finally, to obtain (3.1.14) for some m ≥ 1, we must
6. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 12
have Tn+k
(x)=0(hm+1
). From (2.30) and (2.31), this
will be true if and only if Ci
=0,i=0,1,2…m.
This proves the conditions (2.21) and completes
the proof.
Theorem2.[17-20]
Assumetheconsistencycondition
of (2.10), then the linear multistep method (1.2) is
stable if and only if the following root conditions
(2.11)-(2.12) are satisfied
The root |rj
| 1 =j = 0,1,…,k…(2.33)
|rj
|=1⟹ρ1
(rj
) ≠ 0… (2.34)
Where
ρ α
r r r
k
j
k
j
j
( ) = −
−
=
∑
1
0
Proof:
Given the linear multistep
x x h f a t n
n k
j
k
j n j
k
k
j n j n j
+
=
−
+
=
−
+ +
+ + ≤ ≤ …
∑ ∑
0
1
0
1
α β ; (2.35)
With the associated characteristic polynomial
P r r r
k
j
k
j
j
( ) = − …
+
=
∑
1
0
α (2.36)
such that P(1)=0 by the consistency condition. Let
r0
,…,rn
denote the respective roots of P(r), repeated
according to their multiplying and let r0
=1.
The linear multistep method 2.8. Satisfies the root
condition if
|rj
| ≤ 1,j=0,1,…,k… (2.37)
|rj
|=1⟹P1
(rj
) ≠ 0… (2.38)
Let (2.8) be stable we now prove that the root
condition (2.37) and (2.38) are satisfied. By
contradiction let
|rj
(0)|1 for some j. This is to say we consider the
initial value problem x^1≡0;x(0)=0 with solution
x(t)=0. So that (2.8) becomes
x x n k
n k
j
k
j n j
+
=
−
+
= ≥ …
∑
0
1
α ; (2.39)
If we take x0
=x1
=...=xk
=0, then the numerical
solution clearly becomes xn
=0 for all n ≥ 0.
For perturbed initial values, let
z0
=∈,z1
=∈r1
(0),…,zn
=∈ r1
(0)p
…(2.40)
And for these initial values
max ( )
0
1 0
≤ −
− ≤∈
n k
n n
p
x z r
Which is a uniform bound for all small values of
h, since the right side is independent of h, as ∈→0,
the bound also tend to zero.
The solution (2.12)[22-24]
with the initial condition
(2.13) is simply zn
=∈rj
(0)n
;n ≥ 0. For the derivation
from {xn
}
max ( )
0≤ −
− = → ∞
n k
n n
x z N h
Moreover, the bound that the method is unstable
when |rj
(0)| 0. Hence, if the method is stable, the
root condition |rj
(0)|≤1 must be satisfied.
Conversely, assume the root condition is satisfied,
we now prove for stability restricted to the
exponential equation.
x1
=λx; x(0)=1. (2.41)
This[25-27]
involves solution of non-homogenous
linear difference equations which we simplify by
assuming the roots rj
(0);j=0,1,…,k to be distinct.
The same will be true of rj
(hλ) provided the
values of h is kept sufficiently small, say 0≤ h
≤ h0
. Assume {xn
} and {zn
} to be two solutions of
1 0 1
1 1
0
1
−
( ) − + = ≥ …
+ + +
=
−
+
∑
h x h x n
k n k
j
k
j j n j
λβ α λβ
( ) ;
(2.42)
On (2.10) on [x0
,b] and assume that
max ,
0
0
0
≤ −
− ≤∈ ≤ ≤
n k
n n
x z h h
Introduce the error en
= xn
– zn
and subtracting
using (2.3.8) for each solution
1 0 1
−
( ) − +
( ) = ≤ ≤ …
+ + +
h e h e x x b
k n k j j j k n k
λβ α λβ ;
(2.43)
The general equation becomes
e r h n
n
j
k
j j
n
= ≥
=
∑
1
0
γ λ
( ) ; (2.44)
The coefficient γ0
,…,γk
must be chosen so that
the solution (2.17) will then agree with the given
initial perturbations e0
,…,ek
and will satisfy the
difference equation (2.16). Using the bound
z0
=∈,z1
=∈r1
(0),…,zn
=∈rj
(0)p
and the theory of
linear system of equations, we have
max ;
0
0
0
≤ −
≤ ≤ ≤ …
n k
n c h h
γ ε (2.45)
for some constants cj
0.
To bound the solution en
on [x0
,b], we must bound
each term [rj
(hλ)]n
to do so, consider the expansion
7. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 13
(U) =rj
(0)+Urj
(ξ)…(2.46)
For some ξ between 0 and U. To compute rj
1
(u),
differentiate the identity
p r u u r u
j j
( )
( )− ( )
( )=
σ 0
p r cu r u u u r u r u
j j j j
1 1 1 1
0
( )
( )− ( )− ( )+ ( )
( ) ( )
( )
=
σ σ
r u p r u u r u r u
j j j j
1 1 1
( ) ( )
( )− ( )
( )= ( )
[ ( )
σ σ
r u
r u
p r u ur r u
j
j
1
1
( ) =
( )
( )
( )
( )− ( )
…
σ
( )
(2.47)
By assumption that rj
(0) is a simple root of
p(r)=0; 0 ≤ j ≤ k, it follows that p1
(rj
(0))=0 and
by continuity, p1
(rj
(u))≠0 for all sufficiently small
values of u, the denominator in (2.20) is non-zero
and we can bound rj
(u)|rj
(u)|=c2
for all |u|≤ u0
For
some U0
≥0.
Using this with (3.3.12) and the root condition
(3.2.4), we [Stetter;[28]
Stetter[29]]
have
[ ]
r h r c h c h
j
n
j
λ λ λ
( ) ≤ ( ) + ( ) ≤ + ( )
0 1
2 2
[ ] [ ] (
r h c h e e bx
j
n n C c
n
h h
λ λ λ
( ) ≤ + ( ) ≤ ≤
1 2
2 2
) | λ |
for all 0 ≤ h ≤h0
.
Combine this with (2.18) and (2.19) to get Max|en
|
≤ c2
≤ |ε| ec2
(bxn
) |λ| for an approximate constant
c0
. This concludes the proof.
Theorem2.3:[21-24]
The linear multistep method
(1.2) is said to be convergent if and only if it is
consistent and stable.
Proof:
By this, we want proof that if the consistency
condition is assumed, the linear multistep method
x x h f a t n
n k
j
k
j n j
j
k
j n j n j
+
=
−
+
=
+ +
= + ≤ ≤
∑ ∑
0
1
0
α β ; (2.48)
Is convergent if and only if the root conditions
(2.10) and (2.11) are satisfied.
We assume first the root conditions are satisfied
and then show the linear multi step (2.8)
Is convergent. To start, we use the problem x=0,
x(0) =0 with the solution x (t) = 0. Then, the multi-
step method (2.8) becomes
x x n k
n k
j
k
j n j
+
=
−
+
= ≥ …
∑
0
1
α , (2.49)
With x0
,…,xk
Satisfying n (h) = max|xn
|→0 as h→0… (2.50)
Suppose[25-28]
that the not condition is violated, we
will show that (2.10) is not convergent to x(t)=0.
Assume that some |rj
(0)|1 then a satisfactory
solution of (2.11) is
x h r t t b
n j
n
n
= ( ) ≤ ≤ …
[ ] ;
0 0
Condition (2.10) is satisfied since n(h)=|h(rj
(0))|→0 as h→0.
However, the solution (2.11) does not converge.
First
Max x t x h h r t b
n n j
N h
n
( )− = ≤ ≤
[ ( )
( )
0 0
Consider those values of h
b
N h
=
( )
. Then, L’
Hospital’s rule can be used to show that
Lim
b
N
r
N
( )
0 = ∞
Showing that (2.11) does not converge.
Conversely assume the root condition is satisfied
as with theorem 2.2; it is rather difficult to give
a general proof of converge for an arbitrary
differential equation. The present proof is
restricted to the exponential equation (2.14) and
again we assume that the roots rj
=0 are distinct.
To simplify the proof, we will show that the term
γ0
[r0
(λλ)]n
in the solution
x r h
n
j
k
j j
n
=
=
∑
0
γ λ
( )
Will converge to the solution eλt
on [0,b]. The
remaining terms
γj
|rj
(hλ) |n
,j=1,2,…,k are parasitic solution to
converge to zero as h→0. Expand r_0 (hλ) using
Taylor’s theorem,
r0
(hλ)=r0
(0)+hλr0
2
(0)+0(h2
)
From (2.19) r0
2
1
0
1
1
( ) =
( )
( )
σ
ρ
and using this
consistency condition (2.11), this leads to r0
2
(0)=1.
then
r h h h e h
h
0
2 2
1 0 0
λ λ λ
( ) = + + ( )= + ( )
[ ] [ ] [ ]
r h e h e h
n hn n tn
0
2 2
1 0 1 0
λ λ λ
( ) = + ( ) = + ( )
Thus,
max [ ]
0
0 0 0
≤ ≤
( ) = → → …
t b
t
n
n
r h e as h
(2.51)
8. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 14
We[30,31]
must now show that the coefficient γ0
→1
as h→1. The coefficient γ0
,…,γk
satisfy the linear
system
γ0
+γ1
+...+γk
=x0
γ λ γ λ
0 0 1
r h r h x
k k
( )
+…+ ( ) =
[ ]
γ λ γ λ
0 0 2
r h r h x
k
k k
k
( )
+…+ ( ) = …
[ ] (2.52)
The initial values x0
,…,xk
are assumed to satisfy
r h e x ash
j
n k
»t
n
n
( ) max
0
0 0
≤ −
− → →
However, this implies
Lim xn
=1,0 ≤ n ≤ p…(2.53)
The coefficient γ0
can be obtained using Cramer’s
rule to solve (3.3.5) then
γ0
0
1 1
0 1
1
1 1
1 1 1
=
x
x r r
x r r
r r r
r r r
k
k k k
k
k
k
k k
k
k
The denominator converges to the vandermonde
determinant for r0
(0)=1,r1
(0),…,rk
(0); and this is
non-zero since the roots are distinct. Using (2.13),
the numerator converges to the same quantity as
h→0. Therefore, γ→1 as h→0, using this, along
with (2.10), the solution {xn
} converges to x(t)=eλt
on [0,b]. This completes the proof.
ILLUSTRATIVE EXPERIMENT
Example 3.1 (problem statement)
Write a general-purpose function named
HAMMING that solves a system offirst-order
ordinary differential equations.
dy
dx
f x y y yn
1
1 1 2
= …
( )
, , , , ,
dy
dx
f x y y yn
2
2 1 2
= …
( )
, , , , (3.1)
dy
dx
f x y y y
n
n n
= …
( )
, , , , ,
1 2
Using Hamming’s predictor-corrector method
embedded in C1
and C2
of page 6, Section 2 and
also in,[3]
write a main program that solves the
second-order ordinary differential equation.
d y
dx
y
2
2
= − , (3.2)
Subject to the initial conditions
y
dy
dx
0 0 0 1
( ) = ( ) = (3.3)
The program should call on the fourth-order
Runge-kutta function RUNGE, see[3]
to find
the essential starting values for Hamming’s
algorithm as
y Vo
1 0
, = (3.4)
y2 0 0
, =
And thereafter, should call on HAMMING to
calculate estimates of y and dy/dx on the interval
[0,xmax
]. Evaluate the numerical solutions for
several different step sizes, h, and compare the
numerical results with the true solutions.
y x x
( ) = sin (3.5)
dy
dx
x
= cos .
Method of Solution
Let yj,i
be the final modification of the estimated
solution for the j th dependent variable, yj
,xi
,
resulting from Hamming’s method and left fj,i
be
the calculated estimate of fj
at xi
, that is,
y y x
j i j i
, ,
= ( )
f f x y y y
j i j i i i n i
, , , ,
, , , , .
= …
( )
1 2 (3.6)
Assume that yj,i
,yj,i-1
,yj,i-2
,yj,i-3
, fj,i
,fj,i-1
, fj,1-2,
have
already been found and are available for j=1,2,…,
n. Let an estimate of the local truncation error for
the corrector equation (3.4) for the j th dependent
variable be denoted ej,i
. Then, assuming that
the ej,i
, j=1,2,…,n, are available, the procedure
outlined for Hamming’s method in the above may
be modified to handle a system of n simultaneous
first-order ordinary differential equations by
simply appending a leading subscript, j, to all if all
y and f terms in (3.1) to (3.6). In terms of the new
nomenclature, Steps 2 through 6 of the outline in
the above are:
9. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 15
2. The predicted solutions are computed using
the Milne predictor below;
y y hf f f f j n
j i j i j i j i j i
, , , , , ,
( ), , , ,
+ − − −
= + − + = …
1 0 3 1 2
4
3
2 2 1 2
(3.7)
3. The predicted solutions, yj,i+1,0
, are modified
(assuming that the local truncation error
estimates, ej,i+1
, j=1,2,…,n, will not be
significantly different from estimates ej,i
,
j=1,2,…,n) as in (3.5):
Y y e j n
j i j i j i
, , , , , , , , , .
+ +
= + = …
1 0 1 0
112
9
1 2 (3.8)
4. The j th corrector equation corresponding to
(3.4) is applied for each dependent variable:
y y y h f f
j
j i j i j i j i j i
, , , , , , , ,
, ,..
+ − + −
= + − + +
( )
=
11 2 1 0 1
1
8
9 3 2
1 2 .
.,n
(3.9)
Where
f f x y y y
j i i i i n i
, , , , , , , ,
( , , , , )
+ + + + +
= …
1 0 1 1 1 0 2 1 0 1 0 (3.10)
The corrector in (3.5) is being applied just once
for each variable, the customary practice. The
corrector equations could, however, be applied
more than once, as in (3.6). Note that the subscript
j, in (3.6) is an iteration counter, and is not the index
j, on the dependent variables used throughout this
example.
5. Estimate the local truncation error for each of
the corrector equations on the current interval
as in (3.6)
e y y j n
j i j i j i
, , , , , , , , , .
+ + +
= −
( ) = …
1 11 1 0
9
121
1 2 (3.11)
6. Make the final modifications of the solutions
of the corrector equations as in (3.8)
y y e j n
j i j i j i
, , , , , , , ,
+ + +
= − = …
1 11 1 1 2 (3.12)
After evaluation the yj,i+1,
the n values fj,i+1
may be
computed from (3.13) as
f f x y y y , j n
j i j i i i n i
, , , , ,
, , , , , , .
+ + + + +
= …
( ) = …
1 1 1 1 2 1 1 1 2
(3.13)
If desired, the entire process may be repeated
for the next interval by starting again at Step 2.
Therefore, the function HAMING can be written
to solving arbitrary system of first-order equations,
provided that the 5n+2 essential numbers, xi,h and
yj,i-3
,
f
f
f
e
J n
j i
j i
j i
j i
,
,
,
,
,
,
,
,
, , , ,
−
−
= …
1
2
1 2 (3.14)
Are available for the function to use on entry to
the predictor section, and the essential numbers
and
y
y
y
f
f
f
j n
j i
j i
j i
j i
j i
j i
,
,
, , ,
,
,
,
,
,
,
,
,
, , , ,
−
+
−
−
= …
2
1 0
1
2
1 2 (3.15)
Then, write the calling program and function
HAMMING, so that the matrices Y and F, and
the vector, e ́have the following contents at the
indicated points in the algorithm.
Before entry into the corrector section of
HAMMING:
Y is unchanged from (3.16)
F
f
f
f
f
f
f
f
f
f
i
i
i
i
i
i
j i
j i
j i
=
+
−
+
−
+
−
1 0
1
1 1
2 1 0
2
2 1
1 0
1
,
,
,
, ,
,
,
, ,
,
,
f
f
f
n i
n i
n i
, ,
,
,
+
−
1 0
1
(3.16)
e ́ is unchanged from (3.16)
After return from the corrector section of
HAMMING:
Y
y
y
y
y
y
y
y
y
y
y
i
i
i
i
i
i
j i
j
=
+
−
−
+
−
−
+
1 1
1
1 1
1 2
2 1
2
2 1 1
2 1 2
1
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
i
j i
j i
n i
n i
n i
n i
y
y
y
y
y
y
−
−
+
−
−
1
2
1
1
2
(3.17)
F is unchanged from (3.20)
′ =
e e e e e
i i j i n i
1 2
, , , ,
(3.18)
If the calling program uses (3.14) to replace the
elements in the first row of F after the return from
the corrector section of HAMING, so that
F
f
f
f
f
f
f
f
f
f
f
i
i
i
i
i
i
j i
j i
j i
n
=
+
−
+
−
+
−
1 1
1
1 1
2 1
2
2 1
1
1
,
,
,
,
,
,
,
,
,
,
,
,
,
i
n i
n i
f
f
+
−
1
1
(3.19)
Then, the matrices Y and F and the vector e ́ are
readyforacallonthepredictorsectionofHAMING
for the next integration step. The independent
variable is incremented in the predictor section,
that is,
x x h
i i
+ = +
1 (3.20)
10. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 16
So that the calling program will automatically
have the proper value to calculate the fj,i+1,0
and
fj,i+1
from (3.10) and (3.14), respectively.
Assuming that the integration process begins at
x=x0
and that the only known conditions on (3.2)
are
y1,0
=c1
y2,0
=c2
(3.21)
y c
n n
,0 =
There is not enough information to calculate the
elements of (3.17), (3.18), and (3.19). Therefore,
HAMMING cannot be called directly when x1
=x0
.
The usual procedure is to use a one-step method to
integrate across the first steps to evaluate
y
y
y
j n
j
j
j
,
,
,
, , ,
1
2
3
1 2
= … (3.22)
With (3.15) and (3.16), the matrix Y of (3.17) is
known for i=3. The matrix F of (3.19) may be
evaluated from (3.15) and (3.16) for i=3. The
vector of local truncation error estimates (3.14)
for i=3 is normally unknown, and should be set to
zero unless better values are available from other
sources. Hamming may then be called for the first
time.
In the calling program that follows, the function
RUNGE, already developed in,[3]
is used to
generate the solutions of (3.22). Since RUNGE
implements the fourth-order Runge-kutta method,
the solutions of (3.27) should be comparable in
accuracy with the solutions generated by the
Hamming’s predictor-corrector algorithm, also a
fourth-order method.
The main program reads data values for
n,x0
,h,xmax
,int,y1,0
,y2,0
,…,yn,0
. Here, int is the
number of intergration steps between the printings
of solution values. This program is a reasonably
general one. However, the defining statements
for computation of the derivative estimates (3.11)
and (3.14) would be different for each system of
differential equations.
For test purposes, the differential equation solved
is (3.2), subject to the initial conditions of (3.3).
First, the second-order equation must be written as
a system or two first-order equations. Let
y1
=y
y
dy
dx
2 = (3.23)
Then,
dy
dx
f x y y
dy
dx
y
1
1 1 2 2
= ( )= =
, , (3.24)
dy
dx
f x y y
d y
dx
y y
2
2 1 2
2
2 1
= ( )= = − = −
, ,
So that
f1,i
=y2,i
f2
= –y2,I
(3.25)
The initial conditions of (3.3)
y1,0
=0
y2,0
=1(3.26)
With a known program listing, given, we use the
listed data below
Data
X= 0.0000 H= 1.000000 XMAX= 5.0000
INT= 1 N= 2
YR (1) YR (2)= 0.00000 1.00000
X= 0.0000 H= 0.500000 XMAX= 5.0000
INT= 1 N= 2
YR (1) YR (2) 0.00000 1.00000
X= 0.00000 H= 0.250000 XMAX= 5.00000
INT= 2 N= 2
YR (1) YR (2)= 0.00000 1.00000
X= 0.0000 H= 0.125000 XMAX= 5.0000
INT= 4 N= 2
YR (1) YR (2) 0.00000 1.00000
X= 0.0000 H= 0.062500 XMAX= 5.0000
INT= 8 N= 2
YR (1) YR (2)= 0.00000 1.00000
X= 0.0000 H= 0.031250 XMAX= 5.0000
INT= 16 N= 2
YR (1) YR (2) 0.00000 1.00000
X= 0.00000 H= 0.015625 XMAX= 5.00000
INT= 32 N= 2
YR (1) YR (2)= 0.00000 1.00000
Moreover, we have the following results
Computer Output
Results for the first data set
H = 0.100D 01
XMAX = 5.0000
INT = 1
N = 2
X Y (1) Y (2)
11. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 17
X Y (1) Y (2)
0.0 0.0 0.1000000 01
1.0000 0.8333333D 00 0.5416667D 00
2.0000 0.9027778D 00 –0.4010417D 00
3.0000 0.1548032D 00 –0.9695457D 00
4.0000 –0.6638999D 00 –0.6880225D 00
5.0000 –0.1045312D 00 0.1429169D 00
Results for the second data set
H = 0.500D 00
XMAX = 5.0000
INT = 1
N = 2
X Y (1) Y (2)
0.0 0.0 0.1000000D 01
0.5000 0.4791667D 00 0.8776042D 00
1.0000 0.8410373D 00 0.5405884D 00
1.5000 0.9971298D 00 0.7142556D -01
2.0000 0.9104181D 00 –0.4146978D 00
2.5000 0.6001440D 00 –0.8012885D 00
3.0000 0.1422377D 00 –0.9912065D 00
3.5000 –0.3510641D 00 –0.9387887D 00
4.0000 –0.7588721D 00 –0.6554891D 00
4.5000 –0.9810034D 00 –0.2112528D 00
5.0000 –0.9627114D 00 0.2857746D 00
Results for the third data set
H = 0.250D 00
XMAX = 5.0000
INT = 2
N = 2
X Y (1) Y (2)
0.0 0.0 0.1000000D 01
0.5000 0.4791667D 00 0.8776042D 00
1.0000 0.8410373D 00 0.5405884D 00
1.5000 0.9971298D 00 0.7142556D -01
2.0000 0.9104181D 00 –0.4146978D 00
2.5000 0.6001440D 00 –0.8012885D 00
3.0000 0.1422377D 00 –0.9912065D 00
3.5000 –0.3510641D 00 –0.9387887D 00
4.0000 –0.7588721D 00 –0.6554891D 00
4.5000 –0.9810034D 00 –0.2112528D 00
5.0000 –0.9627114D 00 0.2857746D 00
Results for the fourth data set
H = 0.125D 00
XMAX = 5.0000
INT = 4
N = 2
X Y (1) Y (2)
0.0 0.0 00 0.1000000D -01
0.5000 0.4794249D 00 0.8775832D 00
1.0000 0.8414708D 00 0.5403028D 00
1.5000 0.9974948D 00 0.5403028D -01
2.0000 0.9092971D 00 –0.4161466D 00
2.5000 0.5984717D 00 –0.8011433D 00
3.0000 0.1411195D 00 –0.9899918D 00
3.5000 –0.3507835D 00 –0.9364555D 00
4.0000 –0.7568022D 00 –0.6536421D 00
4.5000 –0.9775289D 00 –0.2107944D 00
5.0000 –0.9589223D 00 0.2836631D 00
Results for the fifth data set
H = 0.625D-01
XMAX = 5.0000
INT = 8
N = 2
X Y (1) Y (2)
0.0 0.0 00 0.1000000D -01
0.5000 0.4794255D 00 0.8775826D 00
1.0000 0.8414710D 00 0.5403023D 00
1.5000 0.9974950D 00 0.7073721D -01
2.0000 0.9092974D 00 –0.4161468D 00
2.5000 0.5984721D 00 –0.8011436D 00
3.0000 0.1411200D 00 –0.9899925D 00
3.5000 –0.3507832D 00 –0.9364566D 00
4.0000 –0.7568025D 00 –0.6536436D 00
4.5000 –0.9775301D 00 –0.2107958D 00
5.0000 –0.9589242D 00 0.2836622D 00
Results for the sixth data set
H = 0.312D-01
XMAX = 5.0000
INT = 16
N = 2
X Y (1) Y (2)
0.0 0.0 00 0.1000000D –01
0.5000 0.4794255D 00 0.8775826D 00
1.0000 0.8414710D 00 0.5403023D 00
1.5000 0.9974950D 00 0.7073720D –01
2.0000 0.9092974D 00 –0.4161468D 00
2.5000 0.5984721D 00 –0.8011436D 00
3.0000 0.1411200D 00 –0.9899925D 00
3.5000 –0.3507832D 00 –0.9364567D 00
4.0000 –0.7568025D 00 –0.6536436D 00
4.5000 –0.9775301D 00 –0.2107958D 00
5.0000 –0.9589243D 00 0.2836622D 00
Results for the seventh data set
H= 0.156D-01
XMAX= 5.0000
INT= 32
N= 2
X Y (1) Y (2)
0.0 0.0 00 0.1000000D –01
0.5000 0.4794255D 00 0.8775826D 00
12. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 18
1.0000 0.8414710D 00 0.5403023D 00
1.5000 0.9974950D 00 0.7073720D –01
2.0000 0.9092974D 00 –0.4161468D 00
2.5000 0.5984721D 00 –0.8011436D 00
3.0000 0.1411200D 00 –0.9899925D 00
3.5000 –0.3507832D 00 –0.9364567D 00
4.0000 –0.7568025D 00 –0.6536436D 00
4.5000 –0.9775301D 00 –0.2107958D 00
5.0000 –0.9589243D 00 0.2836622D 00
DISCUSSION OF RESULTS
Double-precision arithmetic has been used for all
calculations.
Differential equation (3.2) with initial conditions
given by (3.2) has been solved on the interval
[0,5] with step-size h=1.0, 0, 0.25, 0.125, 0.0625,
0.03125, and 0.015625 to seven-place accuracy,
the true solutions.
y1
=y=sinx
y
dy
dx
x
2
= = cos ,(3.27)
Are listed in Tables 1 and 2.
Results for step-sizes 0.03125 and 0.015625 (data
sets 6 and 7) agree with the true values to seven
figures. Results for larger step sizes are not of
acceptable accuracy. The program has been run
with even larger values of h (results not shown)
as well. For h large enough, the solutions “blow
up” in a fashion similar to that already observed
for the Runge-kutta’s method in Example 6.3[29-32]
In view of the periodic nature of the solution
functions, it is not surprising that as the step size
approaches the length of the functional period,
the solutions become meaningless. Clearly, for
step sizes larger than the period, virtually all local
information about the curvature of the function is
lost; it would be unreasonable to expect accurate
solutions in such cases.
CONCLUSION
The findings of the above experiment confirms
that the linear multistep methods are indeed fixed
point iteration methods as stated in the main result.
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Table 1: True solutions,
X Y = sinxdy/dx = cosx
0.0 0.0000000 1.0000000
1.0 0.8414710 0.5403023
2.0 0.9092974 −0.4161468
3.0 0.1411200 −0.9899925
4.0 −0.7568025 −0.6536436
5.0 −0.9589243 0.2836622
Table 2: Calculated solutions at x=5
Step size, hY1
Y2
1.0 −1.045312 0.1429169
0.5 −0.9627114 0.2857746
0.25 −0.9589062 0.2837483
0.125 −0.9589223 0.2836631
0.0625 −0.9589242 0.2836622
0.03125 −0.9589243 0.2836622
0.015625 −0.9589243 0.2836622
True values −0.9589243 0.2836622
13. Emmanuel: Linear multistep methods as fixed point iterative methods
AJMS/Apr-Jun-2022/Vol 6/Issue 2 19
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