The Newton-Raphson Method: A Detailed Analysis

Abstract

The Newton-Raphson method is one of the most popular iterative techniques for solving equations of the form f(x)=0. It is widely used in numerical analysis due to its efficiency and rapid convergence properties when the initial guess is sufficiently close to the true root. This paper explores the underlying theory of the Newton-Raphson method, its mathematical formulation, convergence criteria, and practical applications. Additionally, we examine its advantages and limitations providing insight into the conditions necessary for its optimal performance.

Introduction

I. INTRODUCTION

The Newton-Raphson method, named after Isaac Newton and Joseph Raphson is a powerful iterative method for finding approximate solutions to real-valued functions. Its primary advantage lies in its quadratic convergence rate meaning that the number of correct digits in the approximation roughly doubles with each iteration provided the starting guess is sufficiently close to the true root. The method is widely applicable in fields such as computational mathematics, engineering, physics and economics.

The general problem addressed by the Newton-Raphson method is the equation f(x)=0 where f(x) is a continuous and differentiable function. The goal is to find the root of the function i.e. a value of x for which f(x)=0. This paper examines the method's formulation, convergence behaviour and limitations as well as its practical applications and numerical examples.

II. MATHEMATICAL FORMULATION

A. The Basic Iterative Scheme

The Newton-Raphson method approximates the root of a function by iteratively refining an initial guess, x0, using the following formula:

xn+1 = xn - f(Xn)f'(Xn)

where:

• xn is the current approximation to the root
• f(xn) is the value of the function at xn?
• f′(xn) is the derivative of the function at xn

The method generates a sequence of approximations x0, x1, x2…that, ideally, converges to the true root of the equation.

B. Derivation of the Formula

The Newton-Raphson method is an iterative root-finding algorithm derived using a first-order Taylor series expansion. Consider a function f(x) and a point xn? where we want to approximate the root of f(x)=0.

1) Step 1: First-Order Taylor Expansion

The first-order Taylor expansion of f(x) about xn is:

                                f(x) ≈ f(xn)+f′(xn)(x−xn)

This approximation provides a linear estimate of f(x) near the point xn

2) Step 2: Set f(x)=0

To find the root, we set f(x)=0 (the point where the function intersects the x-axis):

0=f(xn)+f′(xn)(x−xn)

3) Step 3: Solve for x

Solving for x, we get:

                                x = xn - f(Xn)f'(Xn)

This is the Newton-Raphson formula, which gives a new approximation x for the root based on the current approximation xn?, the function value f(xn), and the derivative f′(xn).

4) Step 4: Iterative Process

The method is iterative. To refine the estimate, we update the value of xn in the following way

xn+1 = xn - f(Xn)f'(Xn)

By repeating this process, the sequence {xn} converges to the true root of f(x)=0 if the initial guess is sufficiently close to the root and the function behaves well.

This derivation shows how the Newton-Raphson method uses the first-order approximation of f(x) to iteratively improve the estimate of the roots

III. CONVERGENCE OF THE NEWTON-RAPHSON METHOD

A. Conditions for Convergence

For the Newton-Raphson method to converge to the root, the function f(x) must satisfy certain conditions. These include:

1. Initial Guess: The initial approximation x0? must be sufficiently close to the true root. If the initial guess is too far from the root, the method may fail to converge or may converge to the wrong root.
2. Continuity and Differentiability: The function f(x) must be continuous and differentiable at the root and in a neighbourhood around it. The derivative f ′(x) must not be zero at the root as division by zero would lead to failure of the method.
3. Local Convergence: If the initial guess is close enough to the root and the function behaves "well”, the Newton-Raphson method exhibits quadratic convergence. This means that the error in the approximation decreases roughly quadratically with each iteration.

B. Rate of Convergence

The Newton-Raphson method is renowned for its quadratic convergence when the initial guess is sufficiently close to the true root. This means that the error between the true root xr? and the approximation xn? decreases quadratically with each iteration leading to very rapid convergence.

1) Error Behaviour

Let xr be the true root, and xn be the nth approximation to the root. The error after the nth iteration is given by:

en=?xn−xr?

Quadratic convergence means that the error after the next iteration en+1? is approximately proportional to the square of the current error en?. Mathematically, this can be expressed as:

en+1 ≈ Cen2?

where C is a constant that depends on the function f(x) and its behaviour near the root. This relationship shows that, as the iterations progress, the error decreases very quickly. The number of correct digits in the approximation roughly doubles with each iteration.

2) Implications of Quadratic Convergence

1. Fast Convergence: The quadratic rate of convergence means that the Newton-Raphson method rapidly improves the accuracy of the approximations once the iterations are close to the root.
2. Doubling of Correct Digits: With each iteration, the number of correct digits in the approximation typically doubles. For example, if x0 starts with 1 correct digit, x1? may have 2 correct digits, x2? may have 4 correct digits and so on.

However, this rapid convergence assumes that the initial guess is sufficiently close to the root and that f(x) is well-behaved (i.e., f′(x) is not zero or very small near the root).

Summary: The Newton-Raphson method achieves quadratic convergence, which means that the error en? decreases exponentially with each iteration and the approximation to the root becomes highly accurate very quickly. This characteristic makes the method extremely efficient when the initial guess is close to the true root.

C. Examples of Convergence

Find a root of the equation f(x)=ex−3x2 using the Newton-Raphson method.

Start with an initial guess x0=0.6

Steps:

1) Function and Derivative:

f(x)=ex−3x2          

f ′(x)=ex−6x

2) Iteration Formula

The Newton-Raphson iteration formula is:

xn+1 = xn - f(Xn)f'(Xn)

3) First Iteration

1. • Initial guess: x0=0.6
   • Compute f(x0) and f′(x0)

f(0.6) = e0.6−3(0.6)2 ≈1.8221−1.08 = 0.7421

f ′(0.6) = e0.6−6(0.6) ≈1.8221−3.6=−1.7779

• • Apply the iteration formula:

    x1=0.6 − 0.7421-1.7779  ≈ 0.6+0.4174=1.0174

4) Second Iteration

1. • Initial guess: x0=1.0174
   • Compute f(x0) and f′(x0)

f(1.0174) = e1.0174−3(1.0174)2 ≈ 2.7671−3.1063 = −0.3392

f ′(1.0174) = e1.0174−6(1.0174) ≈ 2.7671−6.1044=−3.3373

• • Apply the iteration formula:

x2=1.0174 − -0.3392-3.3373  ≈ 1.0174−0.1016=0.9158

5) Third Iteration

1. • Initial guess: x0=0.9158
   • Compute f(x0) and f′(x0)

f(0.9158) = e0.9158−3(0.9158)2 ≈ 2.4986−2.5165 = −0.0179

f ′(0.9158) = e0.9158−6(0.9158) ≈ 2.4986−5.4948=−2.9962

• • Apply the iteration formula:

x3=0.9158 − -0.0179-2.9962  ≈ 0.9158−0.0060=0.9098

6) Fourth Iteration

1. • Initial guess: x0=0.9098
   • Compute f(x0) and f′(x0)

f(0.9098) = e0.9098−3(0.9098)2 ≈ 2.4829−2.4825=0.0004

f ′(0.9098) = e0.9098−6(0.9098) ≈ 2.4829−5.4588=−2.9759

Apply the iteration formula:

x4=0.9098 − 0.0004-2.9759  ≈ 0.9098+0.0001=0.9099

Result

After four iterations, we have x4 ≈ 0.9099, which is very close to the actual root of the equation.

Verification

To verify, we can compute f(0.9099):

f(0.9099) = e0.9099−3(0.9099)2 ≈ 2.4831−2.4830 ≈ 0.0001

This small result confirms that x4 ≈ 0.9098 is a very accurate approximation of the root.

Discussion

• The Newton-Raphson method converged quickly to a solution close to the real root.
• This example shows that when the initial guess is reasonable, the method finds highly accurate solutions within a few iterations.

IV. ADVANTAGES OF THE NEWTON-RAPHSON METHOD

A. Fast Convergence

The Newton-Raphson method is known for its rapid convergence, particularly when the initial guess is close to the true root. Its quadratic convergence rate ensures that the number of correct digits increases exponentially with each iteration.

B. Applicability to Nonlinear Equations

The method can be applied to a wide range of nonlinear equations and is often used to solve transcendental equations (e.g. trigonometric, exponential and logarithmic functions) which are difficult to solve analytically.

C. Simplicity of Implementation

The Newton-Raphson method is straightforward to implement requiring only the function and its derivative. For most cases, the method converges quickly making it an efficient choice for solving real-valued equations numerically.

V. LIMITATIONS OF THE NEWTON-RAPHSON METHOD

A. Dependence on Initial Guess

The method is highly sensitive to the initial guess. A poor choice of x0? can lead to divergence, slow convergence or convergence to a local root that is not the desired one. Therefore, the method may require careful consideration of the initial guess or multiple attempts with different starting points.

B. Derivative Calculation

The method requires the calculation of the first derivative of the function. For some functions, this can be computationally expensive or may not be straightforward to obtain. In these cases, derivative-free methods like the Secant method may be more suitable.

C. Failure at Critical Points

If f '(x)=0 at the root or near the root, the method will fail because division by zero occurs. Similarly, if the function has singularities or inflection points close to the root, the method may not converge.

D. Multiple Roots

In cases where the function has multiple roots, the Newton-Raphson method may converge to different roots depending on the initial guess. Special techniques, such as deflation or the use of global search algorithms may be required to find all roots.

VI. APPLICATIONS OF THE NEWTON-RAPHSON METHOD

The Newton-Raphson method is widely used in various scientific and engineering disciplines for solving nonlinear equations. Some key applications include:

1. Root-finding in Engineering: The method is often used in computational fluid dynamics, structural analysis, and control systems to solve nonlinear equations arising from physical models.
2. Optimization: The method is applied in optimization algorithms to find the maximum or minimum of functions, particularly in machine learning and economics.
3. Computational Chemistry: In computational chemistry, the Newton-Raphson method is used to solve nonlinear equations that arise in quantum mechanics and molecular simulations.

VII. ADVANTAGES OF NEWTON-RAPHSON METHOD OVER OTHER METHODS

The Newton-Raphson method offers several distinct advantages over other root-finding methods, particularly in terms of speed, efficiency and applicability to smooth functions. Below are some key advantages:

A. Fast Convergence (Quadratic Convergence)

• Key Advantage: The Newton-Raphson method converges quadratically near the root which means that the number of correct digits roughly doubles with each iteration provided the initial guess is close to the true root.
• Comparison: This is much faster than methods like the Bisection method (linear convergence) and Secant method (superliner convergence) especially for smooth functions.

B. Efficient for Well-Behaved Functions

• Key Advantage: When the function f(x) is smooth and its derivative f′(x) is easily computable, the Newton-Raphson method provides very efficient and fast root-finding.
• Comparison: Other methods like Bisection or False Position are slower because they rely on interval reduction (linear convergence) and cannot take advantage of the function's local behaviour in the same way.

C. No Need for Bracketing

• Key Advantage: Unlike methods such as Bisection and False Position, the Newton-Raphson method does not require an initial bracketing interval. You only need a single initial guess which can be much more convenient.
• Comparison: Bisection and False Position require a bracketing interval with opposite signs at the endpoints which may not always be easy to identify.

D. Direct Use of Derivative Information

• Key Advantage: The Newton-Raphson method directly uses the derivative f′(x) which means that it can incorporate more information about the function's local behaviour to converge more rapidly.
• Comparison: In contrast, methods like Secant approximate the derivative and therefore do not fully utilize the information that an exact derivative could provide. Similarly, Fixed-Point Iteration requires rearranging the equation into a form that may not directly exploit the function’s local properties.

E. Widely Applicable to Smooth Functions

• Key Advantage: When the function f(x) is differentiable, the Newton-Raphson method is highly versatile and can be applied to a wide variety of equations (including transcendental, algebraic and trigonometric equations).
• Comparison: Methods like Fixed-Point Iteration require special rearrangement of the function into a suitable form x=g(x) which may not always be possible. Bisection and False Position are less flexible requiring a sign change between two initial points.

F. Fewer Iterations (When Initial Guess is Good)

• Key Advantage: When a good initial guess is available, fewer iterations are required to achieve a high level of accuracy compared to methods with linear convergence like the Bisection method.
• Comparison: The Secant method may take more iterations than Newton-Raphson to achieve similar accuracy and Bisection will generally be much slower.

G. Works Well for Simple to Moderate Functions

• Key Advantage: For functions where the derivative f′(x) is easy to compute and the function is smooth, Newton-Raphson is highly effective and fast.
• Comparison: Secant method does not require the derivative, but it still generally converges slower than Newton-Raphson. Bisection and False Position methods don't make use of the local information available from the derivative making them slower for smooth functions.

Conclusion

INTRODUCTION The Newton-Raphson method, named after Isaac Newton and Joseph Raphson is a powerful iterative method for finding approximate solutions to real-valued functions. Its primary advantage lies in its quadratic convergence rate meaning that the number of correct digits in the approximation roughly doubles with each iteration provided the starting guess is sufficiently close to the true root. The method is widely applicable in fields such as computational mathematics, engineering, physics and economics. The general problem addressed by the Newton-Raphson method is the equation f(x)=0 where f(x) is a continuous and differentiable function. The goal is to find the root of the function i.e. a value of x for which f(x)=0. This paper examines the method\'s formulation, convergence behaviour and limitations as well as its practical applications and numerical examples. MATHEMATICAL FORMULATION The Basic Iterative Scheme The Newton-Raphson method approximates the root of a function by iteratively refining an initial guess, x0, using the following formula: xn+1 = xn - (f(Xn))/(f^\' (Xn)) where: xn is the current approximation to the root f(xn) is the value of the function at xn f?(xn) is the derivative of the function at xn The method generates a sequence of approximations x0, x1, x2…that, ideally, converges to the true root of the equation. Derivation of the Formula The Newton-Raphson method is an iterative root-finding algorithm derived using a first-order Taylor series expansion. Consider a function f(x) and a point xn where we want to approximate the root of f(x)=0. Step 1: First-Order Taylor Expansion The first-order Taylor expansion of f(x) about xn is: f(x) ? f(xn)+f?(xn)(x?xn) This approximation provides a linear estimate of f(x) near the point xn Step 2: Set f(x)=0 To find the root, we set f(x)=0 (the point where the function intersects the x-axis): 0=f(xn)+f?(xn)(x?xn) Step 3: Solve for x Solving for x, we get: x = xn - (f(Xn))/(f^\' (Xn)) This is the Newton-Raphson formula, which gives a new approximation x for the root based on the current approximation xn, the function value f(xn), and the derivative f?(xn). Step 4: Iterative Process The method is iterative. To refine the estimate, we update the value of xn in the following way xn+1 = xn - (f(Xn))/(f^\' (Xn)) By repeating this process, the sequence {xn} converges to the true root of f(x)=0 if the initial guess is sufficiently close to the root and the function behaves well. This derivation shows how the Newton-Raphson method uses the first-order approximation of f(x) to iteratively improve the estimate of the roots CONVERGENCE OF THE NEWTON-RAPHSON METHOD Conditions for Convergence For the Newton-Raphson method to converge to the root, the function f(x) must satisfy certain conditions. These include: Initial Guess: The initial approximation x0 must be sufficiently close to the true root. If the initial guess is too far from the root, the method may fail to converge or may converge to the wrong root. Continuity and Differentiability: The function f(x) must be continuous and differentiable at the root and in a neighbourhood around it. The derivative f ?(x) must not be zero at the root as division by zero would lead to failure of the method. Local Convergence: If the initial guess is close enough to the root and the function behaves \"well”, the Newton-Raphson method exhibits quadratic convergence. This means that the error in the approximation decreases roughly quadratically with each iteration. Rate of Convergence The Newton-Raphson method is renowned for its quadratic convergence when the initial guess is sufficiently close to the true root. This means that the error between the true root xr and the approximation xn decreases quadratically with each iteration leading to very rapid convergence. Error Behaviour Let xr be the true root, and xn be the nth approximation to the root. The error after the nth iteration is given by: en=?xn?xr? Quadratic convergence means that the error after the next iteration en+1 is approximately proportional to the square of the current error en. Mathematically, this can be expressed as: en+1 ? Cen2 where C is a constant that depends on the function f(x) and its behaviour near the root. This relationship shows that, as the iterations progress, the error decreases very quickly. The number of correct digits in the approximation roughly doubles with each iteration. Implications of Quadratic Convergence Fast Convergence: The quadratic rate of convergence means that the Newton-Raphson method rapidly improves the accuracy of the approximations once the iterations are close to the root. Doubling of Correct Digits: With each iteration, the number of correct digits in the approximation typically doubles. For example, if x0 starts with 1 correct digit, x1 may have 2 correct digits, x2 may have 4 correct digits and so on. However, this rapid convergence assumes that the initial guess is sufficiently close to the root and that f(x) is well-behaved (i.e., f?(x) is not zero or very small near the root). Summary: The Newton-Raphson method achieves quadratic convergence, which means that the error en decreases exponentially with each iteration and the approximation to the root becomes highly accurate very quickly. This characteristic makes the method extremely efficient when the initial guess is close to the true root. Examples of Convergence Find a root of the equation f(x)=ex?3x2 using the Newton-Raphson method. Start with an initial guess x0=0.6 Steps: Function and Derivative: f(x)=ex?3x2 f ?(x)=ex?6x Iteration Formula The Newton-Raphson iteration formula is: xn+1 = xn - (f(Xn))/(f^\' (Xn)) First Iteration Initial guess: x0=0.6 Compute f(x0) and f?(x0) f(0.6) = e0.6?3(0.6)2 ?1.8221?1.08 = 0.7421 f ?(0.6) = e0.6?6(0.6) ?1.8221?3.6=?1.7779 Apply the iteration formula: x1=0.6 ? 0.7421/(-1.7779) ? 0.6+0.4174=1.0174 Second Iteration Initial guess: x0=1.0174 Compute f(x0) and f?(x0) f(1.0174) = e1.0174?3(1.0174)2 ? 2.7671?3.1063 = ?0.3392 f ?(1.0174) = e1.0174?6(1.0174) ? 2.7671?6.1044=?3.3373 Apply the iteration formula: x2=1.0174 ? (-0.3392)/(-3.3373) ? 1.0174?0.1016=0.9158 Third Iteration Initial guess: x0=0.9158 Compute f(x0) and f?(x0) f(0.9158) = e0.9158?3(0.9158)2 ? 2.4986?2.5165 = ?0.0179 f ?(0.9158) = e0.9158?6(0.9158) ? 2.4986?5.4948=?2.9962 Apply the iteration formula: x3=0.9158 ? (-0.0179)/(-2.9962) ? 0.9158?0.0060=0.9098 Fourth Iteration Initial guess: x0=0.9098 Compute f(x0) and f?(x0) f(0.9098) = e0.9098?3(0.9098)2 ? 2.4829?2.4825=0.0004 f ?(0.9098) = e0.9098?6(0.9098) ? 2.4829?5.4588=?2.9759 Apply the iteration formula: x4=0.9098 ? 0.0004/(-2.9759) ? 0.9098+0.0001=0.9099 Result After four iterations, we have x4 ? 0.9099, which is very close to the actual root of the equation. Verification To verify, we can compute f(0.9099): f(0.9099) = e0.9099?3(0.9099)2 ? 2.4831?2.4830 ? 0.0001 This small result confirms that x4 ? 0.9098 is a very accurate approximation of the root. Discussion The Newton-Raphson method converged quickly to a solution close to the real root. This example shows that when the initial guess is reasonable, the method finds highly accurate solutions within a few iterations. ADVANTAGES OF THE NEWTON-RAPHSON METHOD Fast Convergence The Newton-Raphson method is known for its rapid convergence, particularly when the initial guess is close to the true root. Its quadratic convergence rate ensures that the number of correct digits increases exponentially with each iteration. Applicability to Nonlinear Equations The method can be applied to a wide range of nonlinear equations and is often used to solve transcendental equations (e.g. trigonometric, exponential and logarithmic functions) which are difficult to solve analytically. Simplicity of Implementation The Newton-Raphson method is straightforward to implement requiring only the function and its derivative. For most cases, the method converges quickly making it an efficient choice for solving real-valued equations numerically. LIMITATIONS OF THE NEWTON-RAPHSON METHOD Dependence on Initial Guess The method is highly sensitive to the initial guess. A poor choice of x0 can lead to divergence, slow convergence or convergence to a local root that is not the desired one. Therefore, the method may require careful consideration of the initial guess or multiple attempts with different starting points. Derivative Calculation The method requires the calculation of the first derivative of the function. For some functions, this can be computationally expensive or may not be straightforward to obtain. In these cases, derivative-free methods like the Secant method may be more suitable. Failure at Critical Points If f \'(x)=0 at the root or near the root, the method will fail because division by zero occurs. Similarly, if the function has singularities or inflection points close to the root, the method may not converge. Multiple Roots In cases where the function has multiple roots, the Newton-Raphson method may converge to different roots depending on the initial guess. Special techniques, such as deflation or the use of global search algorithms may be required to find all roots. APPLICATIONS OF THE NEWTON-RAPHSON METHOD The Newton-Raphson method is widely used in various scientific and engineering disciplines for solving nonlinear equations. Some key applications include: Root-finding in Engineering: The method is often used in computational fluid dynamics, structural analysis, and control systems to solve nonlinear equations arising from physical models. Optimization: The method is applied in optimization algorithms to find the maximum or minimum of functions, particularly in machine learning and economics. Computational Chemistry: In computational chemistry, the Newton-Raphson method is used to solve nonlinear equations that arise in quantum mechanics and molecular simulations. ADVANTAGES OF NEWTON-RAPHSON METHOD OVER OTHER METHODS The Newton-Raphson method offers several distinct advantages over other root-finding methods, particularly in terms of speed, efficiency and applicability to smooth functions. Below are some key advantages: Fast Convergence (Quadratic Convergence) Key Advantage: The Newton-Raphson method converges quadratically near the root which means that the number of correct digits roughly doubles with each iteration provided the initial guess is close to the true root. Comparison: This is much faster than methods like the Bisection method (linear convergence) and Secant method (superliner convergence) especially for smooth functions. Efficient for Well-Behaved Functions Key Advantage: When the function f(x) is smooth and its derivative f?(x) is easily computable, the Newton-Raphson method provides very efficient and fast root-finding. Comparison: Other methods like Bisection or False Position are slower because they rely on interval reduction (linear convergence) and cannot take advantage of the function\'s local behaviour in the same way. No Need for Bracketing Key Advantage: Unlike methods such as Bisection and False Position, the Newton-Raphson method does not require an initial bracketing interval. You only need a single initial guess which can be much more convenient. Comparison: Bisection and False Position require a bracketing interval with opposite signs at the endpoints which may not always be easy to identify. Direct Use of Derivative Information Key Advantage: The Newton-Raphson method directly uses the derivative f?(x) which means that it can incorporate more information about the function\'s local behaviour to converge more rapidly. Comparison: In contrast, methods like Secant approximate the derivative and therefore do not fully utilize the information that an exact derivative could provide. Similarly, Fixed-Point Iteration requires rearranging the equation into a form that may not directly exploit the function’s local properties. Widely Applicable to Smooth Functions Key Advantage: When the function f(x) is differentiable, the Newton-Raphson method is highly versatile and can be applied to a wide variety of equations (including transcendental, algebraic and trigonometric equations). Comparison: Methods like Fixed-Point Iteration require special rearrangement of the function into a suitable form x=g(x) which may not always be possible. Bisection and False Position are less flexible requiring a sign change between two initial points. Fewer Iterations (When Initial Guess is Good) Key Advantage: When a good initial guess is available, fewer iterations are required to achieve a high level of accuracy compared to methods with linear convergence like the Bisection method. Comparison: The Secant method may take more iterations than Newton-Raphson to achieve similar accuracy and Bisection will generally be much slower. Works Well for Simple to Moderate Functions Key Advantage: For functions where the derivative f?(x) is easy to compute and the function is smooth, Newton-Raphson is highly effective and fast. Comparison: Secant method does not require the derivative, but it still generally converges slower than Newton-Raphson. Bisection and False Position methods don\'t make use of the local information available from the derivative making them slower for smooth functions.

References

[1] Burden, R. L., & Faires, J. D. (2010). Numerical Analysis (9th ed.). Brooks/Cole. [2] Atkinson, K. E. (1989). An Introduction to Numerical Analysis (2nd ed.). Wiley. [3] Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press. [4] Newman, M. (2014). The Newton-Raphson Method and Its Applications in Numerical Analysis. Wiley.

Copyright

Copyright © 2024 Shashank Gawade, Arvind Kudtarkar, Sukanya Sawant, Harshal Wadekar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.