2/4/2003CSCI 150: Introduction to Computer Science1 Introduction to Computer Science CSCI 150 Section 002 Session 6 Dr. Richard J. Bonneau IONA Technologies

2/4/2003 CSCI 150: Introduction to Computer Science

1

Introduction to Computer ScienceCSCI 150 Section 002

Session 6

Dr. Richard J. BonneauIONA Technologies

[email protected]

rbonneau:

Still under construction – use at your own risk !!!

rbonneau:

Still under construction – use at your own risk !!!


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Today’s Outline

Today’s ‘notices’ Review of Last Session Topics Tonight’s Lecture Topics

– Start of Chapter 3 - “Numerical Computation and a Study of Functions”

» Let Us Calculate Some Numbers» Simple Calculations» Functions» Looping and Study of Functions

– New concepts to be developed» Another Pascal data type: real - for big and small numbers» How to read in and write out and compute with real variables» Multiple input of data with readln» Formatting output data – field widths

Next Class Topics


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Today’s Notices

Office Swords 339 extension 2282 Office hours Tu, Th 4-6:15

Don’t forget homework #2 due today! Graded Homework #1 – a few left over from

last week … pick them up?

Does anyone still need the Free Pascal compiler for your own computer??

– Check out the lecture notes page on the course web site or Let me know - by e-mail! ????


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Summary of Last Session

Pluralization program– 4 cases - two special, two general– needed variable task to differentiate between them– direct plural or generated - either replacement or add-on

Mini Text Editor Example– Concatenation Operator (+) for strings– String expressions to insert/delete/replace: e.g.

text := copy(text ….) + new + copy(text….);

– Looping using the while do language construct– Text functions: length, copy, pos

‘Conversation Program’ Pascal Syntax Summary This is the end of Chapter 2 !!!


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Integers, write(ln) and spacing !

Default behavior of writeln – left justify and use as only as many characters as needed

Can specify field width for displaying integers:

writeln(MyIntegerVar:10);

This will right justify the value of the variable MyIntegerVar in a field 10 characters wide. Need to include space for possible minus sign.

Statement write(MyIntegerVar) will display the value and then let the next writeln or write continue on from that point! You can compose better prompts!

Write(‘Hi !’);Writeln(‘ How are you?’);

will display: Hi! How are you?

1034768-1890665536098

10 34768 -18906

65536098

Good fortables ofvalues

AKA Output Formatting

Also worksfor strings!!


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Outline of Chapter 3

Let Us Calculate Some Numbers Simple Calculations Functions Looping and Study of Functions Searching for The Best Value Storing Information in Arrays Findings Sums, Minima, Maxima Patterns in Programming Putting Things in a Row and a Special

Characteristic of Functions

Tonight’s LectureTopics


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Let’s Calculate With Numbers

Last chapter -used computers to do text manipulation using strings (and integers)

Now we want to use the computer to … COMPUTE!!! With Numbers!!

Two sample numerical problems to solve:

A) Find how to save money for future years– Develop an algorithm– Write program to compute savings and refine it

B) Compute largest cylinder’s volume from its surface area: - a packaging problem - and an optimization problem!


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Simple Calculations

First there were Strings – characters, e.g. ‘Pascal’ Then there were Integers - whole numbers (positive,

negative) - 0, 345, -4534 Now we introduce a new data type Now: real - represents fractions, big numbers, small

numbers - uses scientific notation with an exponent and significant digits

234.56 = 2.3456 X 10 2

significant digits exponent In Free Pascal, Real Numbers can be

– as small as 10 -324 – and as big as 10 +308

– with up to 12-13 significant digits!

Example realConstants:1.23453.141591.e20-4.56e-100.000002345


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Pascal and the real data type To declare real variables in Pascal use:

varsavings, rate, fraction : real;

Can assign constants to reals as in

savings := 12.0; rate := 1.0e-1; fraction := -0.0000456;

Can now also compute with real expressions – similar to integer expressions– using the operators of +, -, * and now / (for divide) – – also parentheses– looks like algebra

x := 1.3 + 2.34 * y; avg := (x1 + x2 + x3)/3.0; or more complicated

z := (1+k/n)*(3.14159-sigma)/(.0004 * h);

AnotherData

Type!!

AnotherData

Type!!


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Example - volume of cylinder

Algorithm:

find radius rfind height hcalculate volume V = r2 hprint V

Now let’s look at the Pascal code for this:

Program CylinderVolume (two versions … CylVol1 and CylVol2) – slight difference …

Note - readln may now have two or more input arguments!!! As long as they are numeric!

r

h

r

h


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Another real variable example: Savings Account Growth

How to compute values of savings after a period of time? Need to use compound interest:

newsavings = old savings + (old savings * interest_rate)

And repeat this over and over during a period of time …

In Pascal - use the same variable to accumulate results

savings := savings + (savings*interest_rate);

See program FindSavings!


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New output twist for Reals !! As we saw in the last program, a new formatting notation to

output reals:

writeln( savings : 6 : 2);

Format Specification!!

– The first number (6) is the field size - the least number of characters to display for the whole result

– The second number (2) is the decimal count - how many places to display after decimal point.

In setting field size, always remember to include chars for the sign (+ or -) and the decimal point!

A very common format specification - 10:2 - 10 total chars, 2 decimal places

+/- X X . y y

2

6 = field size


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Two Hazards in Using Reals First Hazard:

Need to understand order of operations - aka precedence:

Consider a * b + c - what gets done first: + or *??

Rule 1: Operations * / have higher precedence than + - Thus a*b done first and then add c to result

Rule 2: If operators have the same precedence, execute from left to right

– e.g. a * b * c / d same as (((a*b) * c) / d)

When in doubt - use parentheses!! Expressions in parentheses done before others!


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More on Numerical Precision

Second Hazard with Reals

numbers not exact in the computer !!!

E.g. 1/3 = .33333333333333…. (infinite)

But in the computer, only finite number of decimal places, e.g. 11-12… leads to ‘round off errors’ in computation

Also problems in dealing with large numbers and small numbers in the same computation!

See Program ErrorDemo to see evidence of when this can be a problem – simply adding and subtracting the same number from a given number

– Use inputs of 100 & then 1e12, 1e13, 1e14 !!


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Functions

A function takes an input, processes it and returns a result or output

y = F( …. ) e.g. y = sin ( x )

Functions can take any kind of inputs (strings, integers, reals …) and produce any kind of outputs, but we here will concentrate on numerical functions.

Functions which have numerical inputs and numerical outputs are very useful throughout computer applications - business, accounting, engineering, science, ...

Rememberyour mathclasses??


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Some ways to represent a function

an English language description

a formula - as in mathematical notation

a table of function values

a visual graph of function values

a computer program!! - can produce many of the other representations!

r Area1.00 3.14 2.00 12.57 3.00 28.27 4.00 50.27

-

10.00

20.00

30.00

40.00

50.00

60.00

1 2 3 4

Series1

Area = r2

the area of a circle is computed by multiplying the irrational number pi by the radius of the circle multiplied by itself


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Looping and Study of Functions

To show values of a function we could use the Pascal while loop to

– set the function’s input value– compute the function value based on the input– print it out– loop back for another input value

This can give us a lot of data that represents the function

Look at the Double Program – very common model – uses a while loop and real (loop) variables to compute a table of

values for the ‘double’ function

– how to modify to go from 0.0 to 1.0 in steps of .1 ???


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Solving Cylinder Volume Problem

Recall again (from high school math!):Volume of cylinder V= r2 hArea of cylinder A= 2 r2 + 2 r h

Let’s assume that we want to find the largest volume for a fixed surface area, e.g. tin

– say at Fixed Area = 1000 = A

Using algebra (!!) we can solve for V to get:

V = 500 r - r3 So now we have a formula that gives us the Volume as

a function of the radius r Consider: CylVol3 Computes a table of function

values of V as a function of radius r Another program does simple graph Cylgra

r

h??

??


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Solving the Savings ProblemGeneral Problem:

– How much would you need to put away monthly to save a million dollars over 40 years!!

First Algorithm: Compute the final total given a monthly amount

How?– Ask for the monthly amount to be saved– Assume a monthly interest rate of 1% = 12% yearly!

– Loop over 40 years (480 months) doing » Compute new monthly balance from old monthly balance and

the interest rate

– Print final balance after 480 iterations

Let’s look at Saving40 for the full program


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Summary of major topics

Reals in Pascal programs– allow to compute with fractions, decimals and very

large and small numbers

Functions and how to work with them in Pascal

Practical application of reals in two sample problems:

– cylinder volume optimization– saving for retirement!


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Next Session

Thursday - Feb. 6

Homework assignment 4 to be passed out

Continuation of Chapter 3– Finding the best or optimal value of a

function!! Let the computer do the job?

– Study the process of evolving a more complex algorithm

Documents

2/4/2003CSCI 150: Introduction to Computer Science1 Introduction to Computer Science CSCI 150 Section 002 Session 6 Dr. Richard J. Bonneau IONA Technologies