Physics 2660: Fundamentals of Scientific Computing Lecture 13 Notes
•  Only 2 weeks left in the semester, 2 lectures left including this one:
–  26 April – TODAY!
–  3 May
•  Labs: one lab left
–  28 April – THURSDAY!
•  Upcoming homeworks: –  HW12 due Monday 2 May at midnight
–  HW13 due Wednesday 4 May at midnight
–  HW14 due Wednesday 4 May at midnight
2
Notes
•  Final exam is coming:
–  Take-­‐‑home projects
•  3 or 4 problems
–  Like a more involved, longer multipart homework assignment
–  Assigned last week of semester on Tuesday 3 May
–  Due Thursday May 12:
•  electronic copies by 9:00am •  hard-­‐‑copies must be submiUed Thursday 12 May between 08:00-­‐‑10:00 in room 022-­‐‑C, our computer lab
Due Thursday May 12,
electronically by 9:00am
and hard copies between
08:00 and 10:00 in 022-C
3
Notes
•  Office hours reminder:
–  My office hours are in Room 022-­‐‑C (our computer lab) from 3:30-­‐‑5pm on Tuesdays or by appointment
•  NOTE: Today they are CANCELED due to an unfortunate conflict with a faculty meeting L
•  If you want to meet with me at some other time this week SEND ME EMAIL and we’ll work something out!
–  TA office hours, also in Room 022-­‐‑C
•  Mondays 5-­‐‑8pm
•  Tuesdays 5-­‐‑8pm
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Review and Today’s Outline
•  Recall, last time:
–  Comparing two models
–  Tuning a model/theory to best match the data
•  Today:
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More on parameter determination
Manipulating binary data (1’s and 0’s)
Numerical solutions to differential equations
Searching
Sorting
Dynamic memory allocation
Linked lists
Further topics
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Tuning a Model to Best Match Some Data
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Tuning a Model
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Tuning a Model
Can we figure out which model – which values of a and b – the data most favors?
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Probability of Some Observation
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Probability of Multiple Observations
The probability of the collection of data – 3 observations – is just the product of the three individual probabilities
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Probability of Multiple Observations
The probability of the collection of data – k observations – is just the product of the k individual probabilities
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P: The χ2 Likelihood Function
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Minimizing the χ2
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Minimizing the χ2
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Minimizing the χ2
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Minimizing the χ2
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More Powerful Application: An Arbitrary Theory
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FiPing with Gnuplot
P(x;a, b, c) =
1
2π c
2
e
−
(ax−b)2
2c 2
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Assessing the Quality of a Fit
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Assessing the Quality of a Fit
Trivial case
Consequence: If the number of fit parameters is greater than or equal to the number of data points the χ2 is undefined.
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Assessing the Quality of a Fit
P(x;a, b, c) =
1
2π c
2
e
−
(ax−b)2
2c 2
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Assessing the Quality of a Fit
So there is a 90% probability that, if the data were consistent with the model (here a Gaussian-­‐‑like thing with 3 params), the data would have a higher chi2 value. Too good to be true? Why are the points so close to the model? Did the fit procedure cheat in some way? Are the uncertainties over-­‐‑estimated?
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FiPing is Done EVERYWHERE
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Curve FiPing
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Deviations from the Model
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The Pull Distribution
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Bias – Is the Prediction In Accord with the Data?
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Clusters of Data Above/Below
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Clusters of Data Above/Below
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More Testing of Compatibility
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Cumulative Distribution Function
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Cumulative Distribution Function
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Example: PDF and CDF
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Empirical Distribution Function
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Empirical Distribution Function
•  The ECDF is made from “unbinned” data
–  not from a binned histogram
–  use raw measured values
•  Do this by:
1.  say you have N values, xi
2.  sort the N values in order of increasing value
3.  plot each of the N values with xi on the x-­‐‑axis and i/
N on the y-­‐‑axis
•
Now, compare model’s CDF and the data’s ECDF…
35
Testing Compatibility
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Testing Compatibility
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Manipulating Binary Data
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Bits and Bytes
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Flipping Bits
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Bitwise Operators
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Bitwise AND
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Bitwise OR
43
Bitwise Left Shift
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Bitwise Right Shift
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Bitwise Inverse
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Bitwise Shift on Constants
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Testing and SePing Bits
(
)
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Testing and SePing Bits: Masks
“Mask”
(
)
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Testing and SePing Bits: Masks
(
)
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Clearing a Bit
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Manipulating Binary Data: Example
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Hexadecimal Representation
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Hexadecimal Representation
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Hexadecimal Representation
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Data Storage
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Example
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Example
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Example
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The Exclusive Or (XOR) Operation
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The Exclusive Or (XOR) Operation
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XOR Application
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XOR Application
Say you want to send this plain-­‐‑text “message”:
1 1 1 0 0 0 0 0 You and the recipient have a copy of some pre-­‐‑
determined “key”
^ ^ ^ ^ ^ ^ ^ ^
1 0 1 0 1 0 1 0 key ^ message = encrypted message:
0 1 0 0 1 0 1 0 And you send this encrypted message to your intended recipient….
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XOR Application
Your recipient receives your encrypted message:
0 1 0 0 1 0 1 0 and applies the key:
^ ^ ^ ^ ^ ^ ^ ^
1 0 1 0 1 0 1 0 1 1 1 0 0 0 0 0 and gets a copy of the plain-­‐‑text message. Successful secret communication!
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XOR Application
One flaw: if someone intercepts both the coded and plain message:
1 1 1 0 0 0 0 0 ^ ^ ^ ^ ^ ^ ^ ^
0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 They can derive the key!
This a problem if the sender/recipient ever re-­‐‑use a key.
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Review: Bitwise Operators
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Review: Testing and SePing Bits
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Storage of Binary Data
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The fputc Function
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The fputc Function
1 byte
4 bytes
1 byte
4 bytes = 4 chars
70
Comparison of ASCII and Binary Storage
10 bytes = 10 chars
4 bytes = 4 chars
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More on Characters
72
Signed v. Unsigned Vars
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Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 10000001b
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Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 10000001b
–  Then
a+b = 1d + (-1d) = ?
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Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 10000001b
–  Then
a+b = 1d + (-1d) = 00000001b + 10000001b
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Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 10000001b
–  Then
a+b = 1d + (-1d) = 00000001b + 10000001b
= 10000010b
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Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 10000001b
–  Then
a+b = 1d + (-1d) = 00000001b + 10000001b
= 10000010b
= -2d
78
Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 10000001b
–  Then
a+b = 1d + (-1d) = 00000001b + 10000001b
= 10000010b
= -2d
–  That looks wrong to me….
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Trick: Two’s Complement Notation
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Trick: Two’s Complement Notation
•  To form the two’s complement of any negative number:
1.  Convert |number| to N-­‐‑bit representation
2.  Subtract |number| from 2N
3.  Add one to the number
•
Examples, 8-­‐‑bit negative numbers:
–
–
If a = -­‐‑1 then |-­‐‑1| in 8-­‐‑bit representation is 00000001
a2c = (11111111 – 00000001) + 00000001
= 11111111
If a = -5 then
a2c = 11111111 – 00000101 + 00000001
= 11111011
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Signed vs. Unsigned Vars
•  Example:
–  Say a = 1d = 00000001b and
b = -1d = 11111111b (using 2’s complement)
–  Then
a+b = 1d + (-1d) = 00000001b + 11111111b
= 00000000b
= 0d
–  That’s beUer!
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More On Binary Data Output
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More On Binary Data Output
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More On Binary Data Output
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More On Binary Data Input
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More On Binary Data Output
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More On Binary Data Output
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More On Binary Data Input
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More On Binary Data Input
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fopen: File I/O Modes
91
fopen, fwrite and fread
92
rewind
93
fseek
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fseek
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ftell
96
feof
97
Writing Binary for Arbitrary Structs
98
Filesize Comparison of Binary v. ASCII Formats
99
Solving Differential Equations
100
A Trajectory
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A Trajectory
We don’t know v(t) but we know v(t0) and the time derivative.
We seek v(t).
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A Trajectory
result:
one step:
not very accurate!!!
All we know is the derivative of v! Here v=y and t=x. 103
A BePer Method
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A BePer Method
All we know is the derivative! 105
A BePer Method
All we know is the derivative! 106
A BePer Method
107
A BePer Method
108
Passing Pointers to Structures
109
Unintended Consequences
110
Unintended Consequences
Warning tells you to rewrite this!
111
Optional Function Arguments
112
Default Argument Values in Functions: C++
113
Default Argument Values in Functions: C++
114
Default Argument Values in Functions: C++
Example:
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Searching in Arrays
116
Searching for a Particular Item
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Method: Linear Search
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Method: Binary Search
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Method: Binary Search
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Method: Binary Search
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Method: Binary Search
122
Method: Binary Search
123
Search Speed: Linear Method
124
Search Speed: Binary Method
125
Sorting
126
Sorting Methods
127
Method: Selection Sort
128
Method: Selection Sort
129
Method: Selection Sort 130
Method: Quicksort 131
Method: Quicksort 132
Method: Quicksort 133
Method: Quicksort 134
Method: Quicksort 135
Method: Quicksort 136