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# Open methods

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### Open methods

1. 1. ROOTS OF EQUATIONS <br />GRAPHIC METHOD<br />
2. 2. GRAPHIC METHOD<br />This method is used primarily for find an interval where function has  any root.<br />Tounderstandbetterwegoingto do anExample.<br />fx=arctan(x)+(x−1)<br /> <br />
3. 3. solution<br />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.<br />So, wegraphthis.<br /> <br />
4. 4. Graphictofindtherood<br />g(x)= arctan x<br />h(x)= 1-x<br />Wherewe can seecrearlythatanintervalwheretherootisonliinterval (0,1)<br />
5. 5. Someconsiderations<br />If fa∗fb<0,  is probably to find an odd number of roots for this equation.<br /> <br />If fa∗fb>0,  is probably to find an even number of roots or that there aren’t.<br /> <br />
6. 6. OPEN METHODS <br />FIXED POINT<br />
7. 7. FIXED POINT<br />This methodisappliedtosolveequations of theformx=gx<br />Iftheequationisfx=0, thenyou can eithercleared x oradded x in bothsides of theequationtoput in theproperly.<br /> <br />CLOSE METHODS<br />
8. 8. Graphicallywehave..<br />f1x=x , f2=g(x)<br /> <br />y=f(x)<br />Raíz<br />Iteratively<br />x1+1=g(xi)<br /> <br />-<br />CLOSE METHODS<br />
9. 9. Someconditionstoconvergence<br />If f2(x)≤f1x  y  f2x>0:  we’llhave a monotonicallyconvergentsolutionbecauseeachsolutionisobtainedclosertotheroot.<br />If f2(x)≤f1x  y  f2x<0:  it has a convergentoscillatorysolutionbecauseeachsolutionisobtained in a mannerclosertotherootoscillatory.<br />Iff2(x)≥f1x  y  f2x>0, so it has a divergentsolution.<br /> <br />CLOSE METHODS<br />
10. 10. example<br />We needfindtherootsforthisequiation:<br />fx=e−x−lnx<br />So<br />gx=e−x−lnx+x<br />then you have to do a table like:<br /> <br />Fixedpoint<br />Tolerance = 0.01<br />
11. 11. CLOSE METHODS <br />NEWTON-RAPHSON<br />
12. 12. Newton-Raphson<br />This method, which is an iterative method is one of the most used and effective.<br /> 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),<br /> <br />f(x)<br />tangente<br />this line intersects the axis x, at a point xi+1that will be our next approximation to the root xr.<br /> <br />Xi<br />Xr<br />Xi+1<br />
13. 13. To calculatedthepointxi+1, first we have to find equation of the tangent line. We know that id has pending<br />m=f′(xi)<br />So, theequationis:<br />y−fxi=f′xix−xi<br />Afterwe do Y=0<br />−fxi=f′xix−xi<br />And solvefor x:<br />x=xi−f(xi)f′(xi)thisistheiterativeformto Newton – Rapson.<br /> <br />Newton-Raphson<br />
14. 14. Newton-Raphson<br />To apreciatedbetterwegonnasolveranexample.<br />fx=e−x−ln(x)<br /> , xo=1<br />tol= 0.01<br /> <br />
15. 15. REFERENCES AND BIBLIOGRAPHY<br /><ul><li>http://noosfera.indivia.net/metodos.html
16. 16. METODOS NUMERICOS PhD EDUARDO CARRILLO – UNIVERSIDAD INDUSTRIAL DE SANTANDER 2010
17. 17. CHAPRA, Steven C. “Métodos Numéricos para Ingenieros”. Edit. McGraw Hil.</li>