Chem 302 Lab Assignment

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Chem 302 Lab Assignment

• Tuesday 230 400ColinChris MJohnathan
  PCraigJonathan EAlex

• Tuesday 400 530CurtisMattChris
  LJanetCoryKristen

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Chem 302 - Math 252

• Chapter 1Solutions of nonlinear equations

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Roots of Nonlinear Equations

• Many problems in chemistry involve nonlinear
  equations
• Linear and quadratic equations are trivial, can
  be solved analytically
• Cubic and higher order solve numerically
• Present a overview of basic methods
• Not a complete discussion

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Successive Approximations

• Simplest method
• Want to solve f(x)0rearrange into the
  form xg(x)
• Use as iteration formula xi1g(xi)
• Use initial guess and iterate until self
  consistent

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Successive Approximations
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Successive Approximations

• Second root

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Successive Approximations

• Second root

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Successive Approximations

• Different formula

Will find x2, will not find x1
Different formula different results No one
formula is best Slow to converge
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Successive Approximations

• How to find initial guesses?
• Grid search

Course to start Get finer
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Analysis of Convergence

• Each stage of iteration xi1g(xi)
• For convergence

• True root

• Intersection of two functions x g(x)

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Analysis of Convergence
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Analysis of Convergence

• Four possibilities
• Monotonic convergence
• Oscillating convergence
• Monotonic divergence
• Oscillating divergence

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Monotonic convergence
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Oscillating convergence
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Monotonic divergence
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Oscillating divergence
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Analysis of Convergence

• Key is g?(x)
• Mean Value Theorem
• If g(x) and g?(x) are continuous on the interval
  a,b then there exists an e (alteltb) such that

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Analysis of Convergence
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Analysis of Convergence
If M lt 1 guaranteed to converge
Sufficient but not necessary
To converge LHS ? 0
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Speed of Convergence
Taylor expansion about root
For xi
When close to convergence

• Dominant term will be 1st nonzero derivative
• Order of Convergence

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Newton-Raphson Method

• One of most common methods
• Usually 2nd order convergence
• Generally superior to simple iteration
• Uses function and 1st derivative to predict root
  (assume f is linear)

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Newton-Raphson Method
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Newton-Raphson Method

• Fairly robust
• Need analytic expressions for f(x) and f'(x)
• May be complicated or not available
• Need to evaluate f(x) and f'(x) many times
• Maximum efficiency
• f'(x) close to zero will cause problems
• Especially important for multiple roots
• Need to checks in program
• Value of f'(x)
• Max iterations

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Newton-Raphson Method

• Test conditions

Convergence
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Secant Method

• NR but use numeric derivative

• Need two points to start

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False-Position Method

• Similar to Secant Method
• Uses two points (one on each side of root) (need
  search)
• Find where function would be zero if linear
  between two points
• During each iteration one point is held fixed
  (pivot), other is moved
• More stable but slower than NR
• If pivot far from root then slow
• If pivot close to root then denominator can be
  small

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False-Position Method

• Algorithm
• Pick xL xR (xL lt lt xR)
• Evaluate
• Calculate
• Calculate
• If then xM is the root
• Replace xL or xR with xM (depends on sign of
  )

xM
xL
xR
Repeat
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False-Position Method
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Bisection Method

• Same as FP Method but xM is average of xL xR
• Drawbacks of both FP Bisection
• Two initial guesses on opposite sides of root
• Multiple or close roots a problem
• f always same sign a problem
• Round-off error as xM gets close to root
• Secant, FP Bisection methods do not require
  analtyic expression of f'(x)

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Roots of Polynomials

• Want as efficient an algorithm as possible
• Efficiency of operations
• /
• Power
• Naive method (10 , 4 )
• Most efficient method for polynomial of order m
  requires m additions and m multiplications
• Nesting (Horners Method)

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Horners Method

• For polynomial of degree m

• Divide by (x-r)

• b0 is f(r)

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Horners Method
m , m
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Horners Method
1 is a root
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Horners Method
2 is a root
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Horners Method

• If b0 0 (i.e. r is a root)

• Have factored (x-r) from equation (i.e. reduced
  order, called deflating)
• Continue to find roots of reduced equation

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Horners Method
2 is a root (i.e. a double root of original
equation)
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Birge-Vieta Method

• NR method with f(x) and f'(x) evaluated using
  Horners method
• Once a root is found, reduce order of polynomial

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Birge-Vieta Method
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Birge-Vieta Method
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Birge-Vieta Method
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Birge-Vieta Method
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Birge-Vieta Method
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Example
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Example

• NR

• x0 ? 4.33 ? 3
• x0 gt 4.33 ? 5
• Third root?

Try it!!!
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• BV

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Roots of Polynomials

• What about complex roots?
• Occur in pairs
• Have form a ib a ib
• Roots of quadratic equation f(x) x2 2ax
  a2 b2
• BV method we removed factors of x r.
• Quadratic can be solved analytically therefore
  best to remove quadratic factors

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Lin-Bairstow Method
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Lin-Bairstow Method

• Iteration Scheme

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Lin-Bairstow Method

• Algorithm
• m gt 3 determine quadratic roots, reduce order of
  problem by 2
• m 3 determine linear root then quadratic roots
• m 2 determine quadratic roots
• m 1 determine linear root

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• // Lin Biarstow.cpp
• include "stdafx.h"
• include ltfstreamgt
• include ltconio.hgt
• include ltstdio.hgt
• include ltmath.hgt
• include ltstdlib.hgt
• include ltstringgt
• include ltdos.hgt
• include ltiostreamgt
• using namespace std
• int _tmain(int argc, _TCHAR argv)
• double t1e-8, a, b, c,u,v,du,dv,r1,r2,ep,f,d
  
• int n,i
• FILE logfile

• anew doublen1
• bnew doublen1
• cnew doublen1
• for(i0iltn1i)
• coutltlt"\nInput coefficient a"ltltiltlt" "
• cingtgtai
• fprintf(logfile,"\nlf lfx",a0,a1)
• for(i2iltn1i)fprintf(logfile,"
  lfxd",ai,i)
• coutltlt"\n\nRoots of the polynomial are\n"
• fprintf(logfile,"\n\nRoots are\n")
• while (an0)n--
• // make sure ngt3
• while(ngt3)

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• b0a0ub1vb2
• fc2c2-c1c3
• if(f0)dudv1
• elsedu(b0c3-b1c2)/fdv(c1b1
  -c2b0)/f
• udu
• vdv
• epsqrt(dududvdv)
• duu4v
• if(dlt0) //complex roots
• r1u/2r2sqrt(-d)/2
• coutltltr1ltlt" i"ltltr2ltltendl
• coutltltr1ltlt" - i"ltltr2ltltendl
• fprintf(logfile,"lf ilf\nlf -
  ilf\n",r1,r2,r1,r2)
• else // real roots

• if(n3)
• u0
• bncnanep1
• while(epgtt)
• for(in-1igt0i--)
• biaiubi1
• cibiuci1
• b0a0ub1
• if(c10)du1
• else du-b0/c1
• udu
• epsqrt(dudu)
• coutltltultltendl
• fprintf(logfile,"lf\n",u)
• n--

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• if(dlt0) //complex roots
• r1u/2r2sqrt(-d)/2
• coutltltr1ltlt" i"ltltr2ltltendl
• coutltltr1ltlt" - i"ltltr2ltltendl
• fprintf(logfile,"lf ilf\nlf -
  ilf\n",r1,r2,r1,r2)
• else
• r1u/2sqrt(d)/2
• r2u/2-sqrt(d)/2
• coutltltr1ltltendl
• coutltltr2ltltendl
• fprintf(logfile,"lf\nlf\n",r1,r2)
• else if(n1)
• r1-a0/a1

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• Roots of the polynomial
• -130.000000 120.000000x -2.000000x2
  -9.000000x3 1.000000x4
• Roots are
• 3.972068
• -3.600135
• 7.399477
• 1.228589