gibson:teaching:fall-2012:math445:pf1
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gibson:teaching:fall-2012:math445:pf1 [2012/12/05 11:17] gibson |
gibson:teaching:fall-2012:math445:pf1 [2012/12/05 13:23] (current) gibson |
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| 4. Suppose matrix //A// has //M// rows and //N// cols. Set //B// to //A// with its columns in reversed order. | 4. Suppose matrix //A// has //M// rows and //N// cols. Set //B// to //A// with its columns in reversed order. | ||
| - | 5. Solve system of equations | + | 5. Solve the system of equations |
| <code> | <code> | ||
| Line 40: | Line 40: | ||
| network of websites. | network of websites. | ||
| - | <network-of-links figure here> | + | {{:gibson:teaching:fall-2012:math445:network2.png?direct&300}} |
| 15. Write Matlab code that converts the connectivity matrix //C// to a | 15. Write Matlab code that converts the connectivity matrix //C// to a | ||
| Line 48: | Line 47: | ||
| - | 16. Given //T//, write Matlab code that computes the vector //x// of | + | 16. Given //T//, write Matlab code that computes the vector $x$ of |
| - | probabilities //x(j)// that you'll end up at website //j// after a | + | probabilities $x_j$ that you'll end up at website $j$ after a |
| long night of random websurfing. | long night of random websurfing. | ||
| - | 17. Write an equation for y as a function of x (log-linear, etc) for | + | 17. Write an equation for //y// as a function of //x// for |
| - | the following data plot | + | the following data plot. Bonus: express exponential functions |
| + | as powers of //e// rather than powers of 10. Use $e^{2.3}\approx 10$ | ||
| + | to convert between the two. | ||
| + | |||
| + | {{:gibson:teaching:fall-2012:math445:fig1.png?direct&300}} | ||
| - | <data plot goes here> | ||
| 18. How would you graph the function $y(x) = x^c$, in a way that highlights | 18. How would you graph the function $y(x) = x^c$, in a way that highlights | ||
| - | this functional relationship? | + | this functional relationship? I.e. given vectors $x$ and $y$ satisfying |
| + | $y_i = x_i^c$, what Matlab command should you use to plot $y$ versus $x$? | ||
| Line 70: | Line 73: | ||
| 21. Write an anonymous function that, for an input vector $x = [x_1, ~x_2]$ | 21. Write an anonymous function that, for an input vector $x = [x_1, ~x_2]$ | ||
| - | returns the output vector $f(x) = [4 x_1 x_2, ~\sin(x_1) cos(x_2)]$ | + | returns the output vector $f(x) = [4 x_1 x_2, ~\sin(x_1) \cos(x_2)]$ |
| 22. Convert the following 2nd order ODE to a 1st order system of ODE in | 22. Convert the following 2nd order ODE to a 1st order system of ODE in | ||
| Line 88: | Line 91: | ||
| <latex> dy/dt = v_y </latex> | <latex> dy/dt = v_y </latex> | ||
| - | <latex> d v_y/dt = -g - \mu v_y^2 </latex> | + | <latex> d v_y/dt = -g - \mu v_y |v_y| </latex> |
| where $g = 9.81, ~\mu =0.35$, $y$ represents the vertical position, and | where $g = 9.81, ~\mu =0.35$, $y$ represents the vertical position, and | ||
| $v_y$ represents the vertical velocity. Represent the two free variables | $v_y$ represents the vertical velocity. Represent the two free variables | ||
| - | with the vector $x = [y, v_y]$ and reexpress the two equations above as | + | with the vector $x = [y, ~v_y]$ and reexpress the two equations above as |
| - | an ODE system | + | an ODE system of the form |
| + | |||
| + | $dx/dt = f(x)$ | ||
| - | $dx/dt = [dx_1/dt, dx_2/dt] = f(x)$ | + | Note that both sides of this equation are vectors: $dx/dt = [dx_1/dt, ~dx_2/dt]$ and |
| + | $f(x) = [f_1(x_1, x_2), ~f_2(x_1, x_2)]$. Your job is to find the functions $f_1$ and $f_2$. | ||
| - | Write an anonymous function in Matlab that computes $dx/dt = f(x)$ given a vector $x$, and | + | Write an anonymous function in Matlab that computes $dx/dt = f(x)$ for an input vector $x$, |
| - | then use //ode45// to integrate the system from $t=0$ to $t=100$ | + | and then use //ode45// to integrate this system from $t=0$ to $t=100$ |
| from the initial conditions $x(0) = [y(0), ~v_y(0)] = [0, 0]$. | from the initial conditions $x(0) = [y(0), ~v_y(0)] = [0, 0]$. | ||
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| <latex> | <latex> | ||
| - | y_i = \sum_{j=1} A_{ij} x_j | + | y_i = \sum_{j=1}^N A_{ij} x_j |
| </latex> | </latex> | ||
| - | for each component $y_i$ of the //M// dimensional vector $y$. But don't that | + | for each component $y_i$ of the //M// dimensional vector $y$. But don't code that |
| - | formula directly! Instead start your code with | + | formula directly! Instead start your function with |
| <code> | <code> | ||
| Line 122: | Line 128: | ||
| </code> | </code> | ||
| - | and write the matrix-vector multiplication as a loop over the $K$ nonzero elements | + | and write the matrix-vector multiplication as a loop over the K nonzero elements |
| - | of $A$. | + | of A. |
gibson/teaching/fall-2012/math445/pf1.1354735060.txt.gz · Last modified: 2012/12/05 11:17 by gibson
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