Open methods
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This document discusses various graphic and numeric methods for finding the roots or zeros of equations, including: 1) The graphic method which graphs the functions to find intervals where roots exist, such as finding the intersection point of y=arctan(x) and y=1-x to solve arctan(x)+x-1=0. 2) The fixed point method which is used to solve equations of the form x=g(x) by iteratively computing xn+1=g(xn). 3) Newton's method which iteratively finds better approximations for roots by using the tangent line approximation at each step as xn+1=xn−f(xn)/f'(xn)
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This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
Numarical values highlighted
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
Quadrature
this is document about Gauss quadrature. For more information please visit http://swiftmemberreview.com
Linear approximations and_differentials
The document discusses linear approximations and differentials. It explains that a linear approximation uses the tangent line at a point to approximate nearby values of a function. The linearization of a function f at a point a is the linear function L(x) = f(a) + f'(a)(x - a). Several examples are provided of finding the linearization of functions and using it to approximate values. Differentials are also introduced, where dy represents the change along the tangent line and ∆y represents the actual change in the function.
Newton raphsonmethod presentation
The Newton-Raphson method ( also known as Newton's method) is a way to quickly find a good approximation for the root of a real-valued function f (x) = 0f (x) = 0. It uses the idea that a nonstop and differentiable function can be approached by a straight line tangent to it.
This slide includes the basic descriptions and function determinations ideas.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear
Mit18 330 s12_chapter4
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.
Similar to Open methods (20)
Numerical Analysis (Solution of Non-Linear Equations) part 2
Numerical Analysis (Solution of Non-Linear Equations) part 2
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...
Open methods
- 1. ROOTS OF EQUATIONS GRAPHIC METHOD
- 2. GRAPHIC METHOD This method is used primarily for find an interval where function has any root. Tounderstandbetterwegoingto do anExample. fx=arctan(x)+(x−1)
- 3. solution To findtherood of f(x), we do arctanx+x−1=0, wherewehavearctanx=1. Thus, theproblemistofindthepoint of intersection of thegraphs of thefunctionsgx=arctanx , y, hx=1−x. So, wegraphthis.
- 4. Graphictofindtherood g(x)= arctan x h(x)= 1-x Wherewe can seecrearlythatanintervalwheretherootisonliinterval (0,1)
- 5. Someconsiderations If fa∗fb<0, is probably to find an odd number of roots for this equation. If fa∗fb>0, is probably to find an even number of roots or that there aren’t.
- 6. OPEN METHODS FIXED POINT
- 7. FIXED POINT This methodisappliedtosolveequations of theformx=gx Iftheequationisfx=0, thenyou can eithercleared x oradded x in bothsides of theequationtoput in theproperly. CLOSE METHODS
- 8. Graphicallywehave.. f1x=x , f2=g(x) y=f(x) Raíz Iteratively x1+1=g(xi) - CLOSE METHODS
- 9. Someconditionstoconvergence If f2(x)≤f1x y f2x>0: we’llhave a monotonicallyconvergentsolutionbecauseeachsolutionisobtainedclosertotheroot. If f2(x)≤f1x y f2x<0: it has a convergentoscillatorysolutionbecauseeachsolutionisobtained in a mannerclosertotherootoscillatory. Iff2(x)≥f1x y f2x>0, so it has a divergentsolution. CLOSE METHODS
- 10. example We needfindtherootsforthisequiation: fx=e−x−lnx So gx=e−x−lnx+x then you have to do a table like: Fixedpoint Tolerance = 0.01
- 12. Newton-Raphson This method, which is an iterative method is one of the most used and effective. Newton-Raphson method does not work on a range bases his formula in an iterative process.Suppose we have the approximation xi to the root xr of f(x), f(x) tangente this line intersects the axis x, at a point xi+1that will be our next approximation to the root xr. Xi Xr Xi+1
- 13. To calculatedthepointxi+1, first we have to find equation of the tangent line. We know that id has pending m=f′(xi) So, theequationis: y−fxi=f′xix−xi Afterwe do Y=0 −fxi=f′xix−xi And solvefor x: x=xi−f(xi)f′(xi)thisistheiterativeformto Newton – Rapson. Newton-Raphson
- 14. Newton-Raphson To apreciatedbetterwegonnasolveranexample. fx=e−x−ln(x) , xo=1 tol= 0.01
- 16. METODOS NUMERICOS PhD EDUARDO CARRILLO – UNIVERSIDAD INDUSTRIAL DE SANTANDER 2010
- 17. CHAPRA, Steven C. “Métodos Numéricos para Ingenieros”. Edit. McGraw Hil.