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39 Cards in this Set
- Front
- Back
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H
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(b-a) / # of intervals
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Error term for trapezoid rule big oh
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O(h^2)
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Error term for simpson's 1/3 rule (big oh)
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O(h^4)
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Error term for simpson's 3/8 rule (big oh)
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O(h^4)
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tarpezoid rule
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(b-a)/2 * (f(a) + f(b))
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Error term for trapezoid rule
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(-1/12) * f''(c) * (b-a) * h^2
exact integration if f'' = 0 |
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composite trapezoid rule
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(b-a) / 2n * (f(a) + 2* f(middle) + 2* f(b))
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Composite tarpezoid rule error
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(-1/12n^2) * (b-a) * h^2 * avg(f''(c))
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Recursive R(0,0)
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1/2(b-a) * (f(a) + f(b))
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Recursive R(n,0)
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(1/2) * R(n-1,0) + h * SUM **k - > 2^n-1** (f(a + (2k-1)h))
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Romberg Integration
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R(n,m) = R(n,m-1) + (1 / (4^m - 1) ) * ((R(n,m-1) - R(n-1,m-1))
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Simpson's 1/3
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(b-a) * ((f(a) + 4*f((b+a)/2)) + f(b))/6)
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Simpson's 1/3 error term
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-h^5 / 90 * f''''(c)
where h = (b-a)/2 |
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Composite Simpson's rule requirements
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Even number of intervals or an odd number of points
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Composite Simpson's Rule
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(b-a) * ((f(a) + 4 * SUM(f of odds) + 2 * SUM(f of evens) + f(b))/3*n)
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Composite Simpson's Rule Error
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(b-a)h^4 / 180 * f''''(c)
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What if we don't have even intervals but want to use simpsons?
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Use simpson's 1/3 on all segements except last and use trapezoidal rule
Use simpson's 3/8 |
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Simpson's 3/8 Rule
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*Can be used on odd intervals*
3/8 * ((b-a)/3) * f(0) + 3*f(x1) + 3*f(x2) + f(x3) |
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Simpson's 3/8 Rule error term
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-3/80 * h^5 * f''''(c)
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Adapative Simpsons (explain and when to stop an interval?)
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(S(2) - S(1)) <= 15*e
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Gaussian Quadature
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SUM(ci * f(xi))
where ci are weights and xi are the nodes |
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Uniform Distribution
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Every number has ssame chance of turning up
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Psuedo-Random
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Computers are limited to 32 bits so repeat eventually happens
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Two common things random number generators have
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Use of large prime numbers and use of modulo arithmetic
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Congruential Method
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I n+ 1 = (a * In + c) mod(m)
Where a,c > 0 and m > all values |
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Multipliciative Method
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Take out the c of congrumential
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Advantage/Disadvantage of congruential?
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Very fast but a poor choice of constants can lead to very poor sequence and the relationship repeats with a period no greater than m (usually aroudn m/4)
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Three properties of pseudo random numbers
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1.) all are periodic
2.) Restriction to limited amount of numbers causes problems in higher dimensions 3.) Individual digits may not be independent (ex. 5 always follows 3 or whatever) |
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Why do we use seeds?
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1.) If we didn't methods would always give same sequence
2.) Make debugging easy |
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Monte Carlo Method
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In Area / (In Area + out of area)
throw random darts *high deviation between samples |
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Monte Carlo Method Error
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1 / sqrt(n)
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Big O for matrix addition for n*n
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O(n^2)
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Big O for matrix multiplication n*n
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O(n^3)
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determinent of a matrix
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1 / (ad-bc) *
[d - b -c a] |
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Upper Form
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All 0's below diagonal
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Lower Form
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All 0's above diagonal
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Back Substitution
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Solve bottom trivially! Remember just plug it in and then go up and use value from below to solve uppers
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Back Substitution big O
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O(n^2) for n*n
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Forward Elimination What is it plus big O
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How we get our matrix into gaussian form
O(n^3) for n*n |
