bisection method for root finding
DESCRIPTION
A numerical method of getting root of an equation is well described.TRANSCRIPT
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CS1201Introduction to Computer Programming II
Rangeet Bhattacharyya
January 14, 2014
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Root nding: bisection
y
xx1x2
x3x1x3x2x3
x2
I You have a function in blue andI two initial points bracketing theroot.
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Root nding: bisection
y
xx1x2
x3
x1x3x2x3
x2
I Find the intersection of the linejoining the two points and zeroaxis.
I Check the sign of f (x3) f (x1).
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Root nding: bisection
y
x
x1
x2
x3
x1
x3x2x3x2
I Get the new bracketing interval.x1 x3
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Root nding: bisection
y
x
x1
x2
x3
x1x3
x2x3x2
I Find the middle of the bracketinginterval and
I check the sign of f (x3) f (x1).
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Root nding: bisection
y
x
x1x2
x3
x1
x3
x2
x3x2
I Get the new bracketing interval.x2 x3
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Root nding: bisection
y
x
x1x2
x3
x1
x3
x2x3
x2
I Find the middle of the bracketinginterval and
I check the sign of f (x3) f (x1)
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Root nding: bisection
y
x
x1x2
x3
x1
x3x2x3
x2
I Get the new bracketing interval.x1 x3
I Stop when |f (xn )| < orwidth of bracket < orcountMaxIter
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Root nding: regula falsi
y
xx1x2
x3x1 x3x1x3x1
I You have a function in blue andI two initial points bracketing theroot.
-
Root nding: regula falsi
y
xx1x2
x3
x1 x3x1x3x1
I Find the intersection of the linejoining the two points and zeroaxis.
I Check the sign of f (x1) f (x3).
-
Root nding: regula falsi
y
x
x1
x2
x3
x1
x3x1x3x1
I Get the new bracketing interval.x1 x3
-
Root nding: regula falsi
y
x
x1
x2
x3
x1 x3
x1x3x1
I Find the intersection andI check the sign of f (x1) f (x3).
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Root nding: regula falsi
y
x
x1
x2
x3x1 x3
x1
x3x1
I Get the new bracketing interval.x1 x3
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Root nding: regula falsi
y
x
x1
x2
x3x1 x3
x1x3
x1
I Find the intersection andI check the sign of f (x1) f (x3)
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Root nding: regula falsi
y
x
x1
x2
x3x1 x3x1x3
x1
I Get the new bracketing interval.x1 x3
I Stop when |f (xn )| < orwidth of bracket < orcountMaxIter
-
Root nding: Secant Method
y
xx1x2
x3x4x5x6
I You have a function in blueand
I two initial points.
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Root nding: Secant Method
y
xx1x2x3
x4x5x6
I Find the intersection of theline joining the two points andzero axis.xn = xn1 yn1 xn2xn1yn2yn1
I You got your next guess.
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Root nding: Secant Method
y
xx1x2x3x4
x5x6
I Get the new guess.xn2 xn1xn1 xnxn = xn1 yn1 xn2xn1yn2yn1
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Root nding: Secant Method
y
xx1x2x3x4x5
x6
I Get the new guess.xn2 xn1xn1 xnxn = xn1 yn1 xn2xn1yn2yn1
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Root nding: Secant Method
y
xx1x2x3x4x5x6
I Get the new guess.xn2 xn1xn1 xnxn = xn1 yn1 xn2xn1yn2yn1
I Stop when |f (xn )| < or countMaxIter
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Root nding: Newton Raphson
y
xx1
x2x3x4x5
I You have a function in blueand
I one initial point.
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Root nding: Newton Raphson
y
xx1x2
x3x4x5
I Find the intersection ofthe tangent at (x , f (x ))and zero axis.xn = xn1 f (xn1)f (x1)
I You got your next guess.
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Root nding: Newton Raphson
y
xx1x2x3
x4x5
I Get the new guess.xn1 xnxn = xn1 f (xn1)f (x1)
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Root nding: Newton Raphson
y
xx1x2x3x4
x5
I Get the new guess.xn1 xnxn = xn1 f (xn1)f (x1)
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Root nding: Newton Raphson
y
xx1x2x3x4x5
I Get the new guess.xn1 xnxn = xn1 f (xn1)f (x1)
I Stop when |f (xn )| < or countMaxIter
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Iterative methods
y
x(x1, 0)
(x1, x2)(x2, x2)
(x2, x3)(x3, x3)
(x3, x4)(x4, x4)
I Start with y = x andy = f (x ).
I Calculate xn = f (xn1).Find (xn , xn ).Let xn1 xn .
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Iterative methods
y
x(x1, 0)
(x1, x2)(x2, x2)
(x2, x3)(x3, x3)
(x3, x4)(x4, x4)
I Start with y = x andy = f (x ).
I Calculate xn = f (xn1).Find (xn , xn ).Let xn1 xn .
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Iterative methods
y
x(x1, 0)
(x1, x2)(x2, x2)
(x2, x3)(x3, x3)
(x3, x4)(x4, x4)
I Start with y = x andy = f (x ).
I Calculate xn = f (xn1).Find (xn , xn ).Let xn1 xn .
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Iterative methods
y
x(x1, 0)
(x1, x2)(x2, x2)
(x2, x3)(x3, x3)
(x3, x4)(x4, x4)
I Start with y = x andy = f (x ).
I Calculate xn = f (xn1).Find (xn , xn ).Let xn1 xn .
I Stop when |xn xn1| < or countMaxIter
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Summary: Root nding
I Why nd roots numerically?I Most equations cant be solved numerically.I For polynomial functions there can be no general solutiona root for polynomials of degree 5 or more.
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Summary: Root nding
I Why nd roots numerically?
I Most equations cant be solved numerically.I For polynomial functions there can be no general solutiona root for polynomials of degree 5 or more.
-
Summary: Root nding
I Why nd roots numerically?I Most equations cant be solved numerically.
I For polynomial functions there can be no general solutiona root for polynomials of degree 5 or more.
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Summary: Root nding
I Why nd roots numerically?I Most equations cant be solved numerically.I For polynomial functions there can be no general solutiona root for polynomials of degree 5 or more.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Merits and demerits of the methods
I Graph plottingEasiest. Not accurate. Precision depends on plotter
I BisectionEasier, always works (for good brackets). Can be slow
I Regular FalsiAlways work (good bracket, no sharp bend). Can be slow
I SecantFast. Need good guesses (no extrema)
I Newton RaphsonFaster. Works for complex numbers May go into traps,need good guess
I Iterative methodsEasy Convergence is not guaranteed.
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).
I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!
-
Root nding: useful tips
I Plotting helps. Visualization often help nd good brackets(or starting point).
I For smooth functions usually Newton Raphson gives goodresult.
I For functions with sharp bend or extrema within thebracket bisection may be better choice.
I Always start by restricting the number of iterations tonite value.
I Check for roots at the ends (of the bracket).I Check for good bracketing interval (for bisection andregula falsi).
I Good luck!