CS1201Introduction to Computer Programming II

Rangeet Bhattacharyya

January 14, 2014

Root nding: bisection

y

xx1x2

x3x1x3x2x3

x2

I You have a function in blue andI two initial points bracketing theroot.

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).

Root nding: bisection

y

x

x1

x2

x3

x1

x3x2x3x2

I Get the new bracketing interval.x1 x3

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).

Root nding: bisection

y

x

x1x2

x3

x1

x3

x2

x3x2

I Get the new bracketing interval.x2 x3

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)

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

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).

Root nding: regula falsi

y

x

x1

x2

x3x1 x3

x1

x3x1

I Get the new bracketing interval.x1 x3

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)

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.

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.

Root nding: Secant Method

y

xx1x2x3x4

x5x6

I Get the new guess.xn2 xn1xn1 xnxn = xn1 yn1 xn2xn1yn2yn1

Root nding: Secant Method

y

xx1x2x3x4x5

x6

I Get the new guess.xn2 xn1xn1 xnxn = xn1 yn1 xn2xn1yn2yn1

Root nding: Secant Method

y

xx1x2x3x4x5x6

I Get the new guess.xn2 xn1xn1 xnxn = xn1 yn1 xn2xn1yn2yn1

I Stop when |f (xn )| < or countMaxIter

Root nding: Newton Raphson

y

xx1

x2x3x4x5

I You have a function in blueand

I one initial point.

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.

Root nding: Newton Raphson

y

xx1x2x3

x4x5

I Get the new guess.xn1 xnxn = xn1 f (xn1)f (x1)

Root nding: Newton Raphson

y

xx1x2x3x4

x5

I Get the new guess.xn1 xnxn = xn1 f (xn1)f (x1)

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

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 .

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 .

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 .

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

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.

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.

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!