Computer Science 121

Scientific Computing Winter 2012 Chapter 1: What is Computation? Computer Science 121

1.1 Computation as Transformation • A computation is a transformation from on or more inputs to an output. • Input = time; Output = reactant (grams)

1.1 Computation as Transformation • Input = age (years); Output = recall (items)

http://www.macalester.edu/~kaplan/scientificprogramming/figures.ppthttp://www.macalester.edu/~kaplan/scientificprogramming/figures.ppt 1.1 Computation as Transformation Input = x; Output =√x

http://www.macalester.edu/~kaplan/scientificprogramming/figures.ppthttp://www.macalester.edu/~kaplan/scientificprogramming/figures.ppt 1.1 Computation as Transformation Input = financial info; Output = tax owed / refunded

http://www.macalester.edu/~kaplan/scientificprogramming/figures.ppthttp://www.macalester.edu/~kaplan/scientificprogramming/figures.ppt 1.1 Computation as Transformation Input = image file; Output = “This is the Mona Lisa!”

1.1 Computation as Transformation Input = .wav file; Output = “This is the vowel /i/”

1.2 Computation as Reaction to Events • Inputs often come from measurements • Measurements are typically performed using transducers (convert input signal into numbers) • microphone • speedometer • thermometer • seismograph • radar • (What sort of data doesn't require transducers?)

1.2 Computation as Reaction to Events • Sundial (and compass): Analog transducers http://research.utk.edu/ora/explore/sundial.jpg

1.2 Computation as Reaction to Events • Because measurements (signals) typically change over time, we get the concept of state: the set of values describing a system at a given time • E.g. • Health: <blood pressure, temperature, heart rate> • Airplane: <heading, altitude, fuel> • Subatomic particle: <position, momentum> • So... • A computation isa transition from an old state to a new state.

1.2 Computation as Reaction to Events • A computation isa transition from an old state to a new state • What does this mean? • If we can build a computational model of how a system changes state (reacts to events), we can make predictions about how the system will behave • Health: influence of drugs • Airplane: response to turbulence • Particle: Heisenberg's Uncertainty Principle (!!!)

1.3 Algorithms Muhammad ibn Mūsā al-Khwārizmī (c. 780 – 850) Al-Kitāb al-mukhtaṣar fī hisāb al-jabr wa-l-muqābala

Matlab Programs Matlab Interpreter Operating System Hardware 1.3 Algorithms • “Don't Reinvent the Wheel” Principle: We should build new computations out of ones that already exist • Principle applies across the board in computer science: • How computer runs programs:

my_program.m read_data.m save_data.m plot_data.m 1.3 Algorithms • Principle applies across the board • How programs are written:

1.3 Algorithms • Build new computations out of ones that already exist • An algorithmis a set of directions for carrying out a computation in terms of other, simpler computations. • Difficulty in writing algorithms comes from differences between the ways that people do things and the way that computers do them • People: Slow, parallel, good generalization

“Cat!” “Cat!” “Cat!” 1.3 Algorithms • People: Slow, parallel, good generalization

6068 x 4859 29484412 1.3 Algorithms • Computers: Fast, serial, poor generalization

Expressing Algorithms • Example: Compare two Arabic numerals (e.g., 23 and 97) to determine which is larger • Count the digits in each numeral. If one has more digits than another, the one with more is the larger. • Otherwise, compare the leftmost digits in the two numerals. If one digit is larger than the other, the numeral with the larger leftmost digit is the larger numeral. • Otherwise, chop off the first numeral and go to step 2. Q.: What haven't we considered?

Expressing Algorithms • Example: Find smallest number in list • Assume smallest is first item in list. • If there are no more items in list, we're done. Report smallest number. • Otherwise, look at next item. If it's smaller than current assumed “smallest” value, make current answer equal to value of this item. • Go to step 2.

Expressing Algorithms • Example: Find smallest LIST: 5 2 9 1 3

Expressing Algorithms • Example: Find smallest LIST: 5 2 9 1 3 SMALLEST: 5

Expressing Algorithms • Example: Sort a list of numbers • Create a new, empty list • Find the smallest item in the old list. • Remove this item from the old list and add it to the beginning of the new list. • If the old list is empty, we're done. Output the new list. • Otherwise, go to step 2.

Expressing Algorithms • Example: Sort a list of numbers OLD: 5 2 9 1 3 NEW:

Expressing Algorithms • Example: Sort a list of numbers OLD: 5 2 9 3 NEW: 1

Expressing Algorithms • Example: Sort a list of numbers OLD: 5 9 3 NEW: 1 2

Expressing Algorithms • Example: Sort a list of numbers OLD: 5 9 NEW: 1 2 3

Expressing Algorithms • Example: Sort a list of numbers OLD: 9 NEW: 1 2 3 5

Expressing Algorithms • Example: Sort a list of numbers OLD: NEW: 1 2 3 5 9

Important Points • We re-used our smallest-number algorithm in our sorting algorithm: instead of re-writing it, we referred to it! • We needed a strange notion of “smallest” - smallest item in a one-item list is the item itself. • Contrast this with ordinary language terminology • No “smallest” (or largest) item in a one-item list. • We (are supposed to) say “smaller” (not “smallest”) if there are only two items • These sorts of differences contribute to our difficulty in learning to program. • But as algorithms get more complicated, we will find it more difficult to use ordinary language (“the current smallest”) to express them – need variables

Computer Science 121