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CIS 541 – Numerical Methods. Mathematical Preliminaries. Derivatives. Recall the limit definition of the first derivative. Partial Derivatives. Same as derivatives, keep each other dimension constant. (1,f’(t)). f(x). t. Tangents and Gradients.

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CIS 541 – Numerical Methods

Mathematical Preliminaries

• Recall the limit definition of the first derivative.

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• Same as derivatives, keep each other dimension constant.

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f(x)

t

• Recall that the slope of a curve (defined as a 1D function) at any point x, is the first derivative of the function.

• That is, the linear approximation to the curve in the neighborhood of t is l(x) = b + f’(t)x

y

x

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• Since we also want this linear approximation to intersect the curve at the point t.

l(t) = f(t) = b + f’(t)t

• Or, b = f(t) - f’(t)t

• We say that the line l(x)interpolates the curve f(x) at the point t.

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• We can think of the curve shown in the previous slide as the set of all points (x,f(x)).

• Then, the tangent vector at any point along the curve is

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• There are other ways to represent curves, rather than explicitly.

• Functions are a subset of curves (x,y(x)).

• Parametric equations represent the curve by the distance walked along the curve (x(t),y(t)).

Circle: (cos, sin)

• Implicit representations define a contour or level-set of the function: f(x,y) = c.

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• In higher-dimensions, we have the same thing:

• A surface is a 2D function in 3D:

Surface = (x, y, f(x,y) )

• A volume or hyper-surface is a 3D function in 4D:

Volume = (x, y, z, f(x,y,z) )

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• The linear approximation to the higher-dimensional function at a point (s,t), has the form: ax+by+cz+d=0, or z(x,y) = …

• What is this plane?

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Images courtesy of TJ Murphy: http://www.math.ou.edu/~tjmurphy/Teaching/2443/TangentPlane/TangentPlane.html

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• The formula for the plane is rather simple:

• z(s,t) = f(s,t) - interpolates

• z(s+dx,t) = f(s,t) + fx(s,t)dx = b + adx

• Linear in dx

• Of course, the plane does not stay close to the surface as you move away from the point (s,t).

• OSU/CIS 541

• The normal to the plane is thus:

• The 2D vector:is called the gradient of the function.

• It represents the direction of maximal change.

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• The gradient thus indicates the direction to walk to get down the hill the fastest.

• Also used in graphics to determine illumination.

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• Extrema of a function occur where f’(x)=0.

• The second derivative determines whether the point is a minimum or maximum.

• The second derivative also gives us an indication of the curvature of the curve. That is, how fast it is oscillating or turning.

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• Specific functions of the form:

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• For many polynomials, the latter coefficients are zero. For example:

p(x) = 3+x2+5x3

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• For a function, f(x), about a point c.

• I.E. A polynomial

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• Taylor’s Theorem allows us to truncate this infinite series:

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• Some things to note:

• (x-c)(n+1) quickly approaches zero if |x-c|<<1

• (x-c)(n+1) increases quickly if |x-c|>>1

• Higher-order derivatives may get smaller (for smooth functions).

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• What is the 100th derivative of sin(x)?

• What is the 100th derivative of sin(3x)?

• Compare 3100 to 100!

• What is the 100th derivative of sin(1000x)?

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• Hence, for points near c we can just drop the error term and we have a good polynomial approximation to the function (again, for points near c).

• Consider the case where (x-c)=0.5

• For n=4, this leads to an error term around 2.6*10-4 f()

• Do this for other values of n.

• Do this for the case (x-c) = 0.1

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• Eight terms for first series not even yielding a single significant digit.

• Only four for second serieswith foursignificantdigits.

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• Special case of Taylor’s Theorem, where n=0, x=b.

• Assumes f(x) is continuous and its first derivative exists everywhere within (a,b).

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• So what!?!?! What does this mean?

• Function can not jump away from current value faster than the derivative will allow.

f(x)

a

secant

b

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• If a and b are roots (f(a)=f(b)=0) of a continuous function f(x), which is not everywhere equal to zero, then f’(t)=0 for some point t in (a,b).

• I.e., What goes up, must come down.

f(x)

f’(t)=0

a

t

b

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• For Taylor’s Series and Taylor’s Theorem to hold, the function and its derivatives must exist within the range you are trying to use it.

• That is, the function does not go to infinity, or have a discontinuity (implies f’(x) does not exist), …

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const int Max_Iters = 100,000,000;

float x = -0.1;

float delta = 0.2 / Max_Iters;

float Reimann_sum = 0.0;

for (int i=0; i<Max_Iters; i++)

{

Reimann_Sum += sinf(x);

x+=delta;

}

Printf(“Integral of sin(x) from –0.1->0.1 equals: %f\n”, Reimann_Sum*delta );

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const int Max_Iters = 100,000,000;

float x = -0.1;

float delta = 0.2 / Max_Iters;

float Reimann_sum = 0.0;

for (int i=0; i<Max_Iters; i++)

{

Reimann_Sum += my_sin(x); //my own sine func

x+=delta;

}

Printf(“Integral of sin(x) from –0.1->0.1 equals: %f\n”, Reimann_Sum*delta );

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my_sin( const float x )

{

float x2 = x*x;

float x3 = x*x2;

return (x – x3/6.0 + x2*x3/120.0 );

}

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Static const float fac3inv = 1.0 / 6.0f;

Static const float fac5inv = 1.0 / 120.0f;

my_sin( const float x )

{

float x2 = x*x;

return x*(1.0 – x2*(fac3inv - x2*fac5inv));

}

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const int Max_Iters = 100,000,000;

float x = -0.1;

float delta = 0.2 / Max_Iters;

float Reimann_sum = 0.0;

for (i=0; i<Max_Iters; i++)

{

x2 = x*x;

Reimann_Sum += x*(1.0–x2*(fac3inv-x2*fac5inv);

x+=delta;

}

Printf(“Integral of sin(x) from –0.1->0.1 equals: %f\n”, Reimann_Sum*delta );

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• Pentium III, 600MHz machine

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• Is the result correct?

• Why did we gain some accuracy with version 2.0?

• Is (–0.1,0.1) a fair range to consider?

• Is the original sinf() function optimized?

• How did we achieve our speed-ups?

• We will re-examine this after Lab1.