Developing Microsoft .Net Applications for Windows (Visual Basic .Net) Class 7

1. COMP3100E Developing Microsoft .Net Applications for Windows (Visual Basic .Net)Class 7

2. Objectives • Homework • Recursion

3. Recursion From Wikipedia, the free encyclopedia. • In mathematics and computer science, recursion specifies (or constructs) a class of objects (or an object from a certain class) by defining a few very simple base cases (often just one), and then defining rules to break down complex cases into simpler cases. • Recursion is sometimes mistaken for circular reasoning. However, the crucial difference is that its base cases are defined in terms that are not part of the system. Since all the cases break down into base cases, and the base cases are in different terms, the analysis stops. That is, it is not circular. • An example of a recursive imageFor example, the following is a recursive definition of person's ancestors: • One's parents are one's ancestors (base case); • The parents of any ancestor are also ancestors of the person under consideration (recursion step). • For instance, your ancestors are: • your parents, and • your parents' parents (= grandparents), and • your grandparents' parents, and • everyone else you get by successively adding ancestors • It is convenient to think that a recursive definition defines objects in terms of "previously defined" objects of the class to define.

4. Recursion • Definitions such as these are often found in mathematics. For example, the formal definition of natural numbers is: 0 is a natural number, and each natural number has a successor, which is also a natural number. • To visualize recursion, it can be helpful to consider recursively-defined geometric figures, such as the Koch curve, the Sierpinski triangle, or the Cantor set. • Also, examples of recursion abound in natural language, often appearing as, or transforming into jokes. These jokes can help one develop an intuition for recursion. • For example, responding to the question, "What do you mean what do I mean?" with "What do you mean 'What do you mean what do I mean?'?", can clearly go on forever, although it is unclear how many iterations can be meaningful... • Although artist- and poet-types often make jokes of this kind, studying recursion in mathematics and programming languages can help to understand the meaning and philosophy of language in this sense, and a careful observation of language in this sense can help to understand recursion.

5. Koch Curve

6. Koch Curve • The Koch curve is a mathematical curve, and one of the earliest fractal curves to have been described. It appeared in a 1904 paper entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire" by the Swedish mathematician Helge von Koch. The better known Koch snowflake (or Koch star) is the same as the curve, except it starts with an equilateral triangle instead of a line segment. Eric Haines has developed the sphereflake fractal, a three-dimensional version of the snowflake. • One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows: • divide the line segment into three segments of equal length. • draw an equilateral triangle that has the middle segment from step one as its base. • remove the line segment that is the base of the triangle from step 2. • After doing this once the result should be a shape similar to the Star of David. • The Koch curve is in the limit approached as the above steps are followed over and over again. • The Koch curve has infinite length because each time the steps above are performed on each line segment of the figure its length increases by one third. The length at step n will therefore be (4/3)n and the fractal dimension is log4/log3 =~1.26 (bigger than the dimension of a line {1} but smaller than Peano's Space-filling curve {2}). • The Koch curve is continuous, but not differentiable anywhere. • The area of the Koch snowflake is 8/5 that of the initial triangle, so an infinite perimeter encloses a finite area.

7. Sierpinski Triangle • The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński. • Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski gasket. • For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge.

8. Recursion in Language • Linguist Noam Chomsky produced evidence that unlimited extension of a language such as English is possible only by the recursive device of embedding sentences in sentences. • Thus, a talky little girl may say, "Dorothy, who met the wicked Witch of the West in Munchkin Land where her wicked Witch sister was killed, liquidated her with a pail of water." • Clearly, two simple sentences — "Dorothy met the Wicked Witch of the West in Munchkin Land" and "Her sister was killed in Munchkin Land" — can be embedded in a third sentence, "Dorothy liquidated her with a pail of water," to obtain a very talky sentence. • Niels K. Jerne, the 1984 Nobel Prize laureate in Medicine and Physiology, used Chomsky's transformational-generative grammar model to explain the human immune system, equating "components of a generative grammar ... with various features of protein structures." The title of Jerne's Stockholm Nobel lecture was The Generative Grammar of the Immune System. • Here is another, perhaps simpler way to understand recursive processes: • Are we done yet? If so, return the results. Without such a termination condition a recursion would go on forever. • If not, simplify the problem, solve those simpler problem(s), and assemble the results into a solution for the original problem. Then return that solution. • A more humorous illustration goes: • "In order to understand recursion, one must first understand recursion." Or perhaps more accurate is the following due to Andrew Plotkin: "If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to Douglas Hofstadter than you are; then ask him or her what recursion is." • Examples of mathematical objects often defined recursively are functions, sets, and especially fractals.

9. Recursive Humor • Recursion • See "Recursion". • This is a parody on references in dictionaries, which in some careless cases may lead to circular definitions; in fact the above is the shortest possible one. • Every joke has an element of wisdom, and also an element of misunderstanding. • This one is also the second-shortest possible example of an erroneous recursive definition of an object, the error being the absence of the termination condition (or lack of the initial state, if to look at it from an opposite point of view). • Newcomers to recursion are often bewildered by its apparent circularity, until they learn to appreciate that a termination condition is key. • Other examples are recursive acronyms, such as GNU, PHP or TTP (Dilbert; "The TTP Project"). • "GNU" is a recursive acronym for GNU's Not Unix. • Today, the official meaning is the recursive acronym "PHP: Hypertext Preprocessor". • TTP stands for “The TTP Project” • The labels on Miller beer say "Miller MGD Genuine Draft". • However, MGD stands for Miller Genuine Draft. • So the labels are saying "Miller Miller Genuine Draft Genuine Draft". • "The way to understand recursion is to understand recursion"

10. Recursion in Programming • Recursion in computer programming defines a function in terms of itself. • One example application of recursion is in parsers for programming languages. • The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program. • One basic form of recursive computer program is to define one or a few base cases, and then define rules to break down other cases into the base case. This is analytic, and is the most common design for parsers for computer languages. • Another, similar form is generative recursion. This is synthetic. • In this scheme, the computer uses rules to assemble cases, and starts by selecting a base case. • This scheme is often used when a computer must design something automatically, such as code, a machine part or some other data.

11. Recursion in Programming • The Quicksort and Mergesort algorithms are also commonly done using recursion, which allows them to run in an average of O(n log n) time. • Many operations involving tree data structures also use recursion, as it is often quite difficult to iteratively traverse a tree of unknown length. • In addition, some numerical methods for finding approximate solutions to mathematical equations use recursion. In Newton's method, for example, an approximate root of a function is provided as initial input to the method. The calculated result (output) is then used as input to the method, with the process repeated until a sufficiently accurate value is obtained.

12. Displaying values from 1 to 10 • The Program Structure Loop is generally used in programming to print the values of one to 10 • Example: Sub oldcount(ByVal max As _ Integer) Dim num As Integer For num = 1 To max LstNumbers.Items.Add(num) Next End Sub

13. Inclusively Recursive Sub newcount(ByVal num As _ Integer) If num > 1 Then newcount(num - 1) End If LstNumbers.Items.Add(num) End Sub

14. The Newtonian Square Root #include <stdio.h> #include <math.h> enum boolean {no, yes}; void main(void) { float oldval, newval; int num; enum boolean finished = no; printf("\nEnter value for square root: "); scanf("%d",&num); oldval= (1.0+num)/2.; while (!finished) { newval=(oldval+num/oldval)/2.; if (newval >= oldval) finished= yes; oldval = newval; } printf("Square root of %d is %f",num,newval);

15. Newton Square Root in VB.Net Private Function NewtonSqrt(ByVal num As Integer) As Double Dim oldval, newval As Double Dim finished As Boolean newval = 0 If num > 0 Then finished = False oldval = (1.0 + num) / 2 Do newval = (oldval + num / oldval) / 2 If (newval >= oldval) Then finished = True End If oldval = newval Loop While Not finished End If Return (newval) End Function

16. Newtonian Recursive Square Root Private Function betterGuess(ByVal num _ As Double, ByVal oldval As Double) As Double Return ((oldval + num / oldval) / 2) End Function Private Function closeEnough(ByVal newval _ As Double, ByVal oldval As Double) As Boolean Return (Math.Abs(newval - oldval) < 0.000001) End Function Private Function RecursiveSqrt(ByVal num _ As Double, ByVal oldval As Double) As Double Dim answer As Double If closeEnough(num / oldval, oldval) Then answer = oldval Else answer = RecursiveSqrt(num,_ betterGuess(num, oldval)) End If Return (answer) End Function

17. Calculating Pi Private Function Pi(ByVal num As Integer) As Double If (num = 1) Then Return (1) Else If (num Mod 2 = 1) Then Return (Pi(num - 1) + 1 / (2 * num - 1)) Else Return (Pi(num - 1) - 1 / (2 * num - 1)) End If End If End Function

18. Calculating Pi Again Private Function PiAgain(ByVal n As Integer) As Double If (n = 1) Then Return (0.833333) Else If (n Mod 2 = 1) Then Return (PiAgain(n - 1) + (Power(0.5, 2 * n - 1) + Power(0.33333, 2 * n - 1)) / (2 * n - 1)) Else Return (PiAgain(n - 1) - (Power(0.5, 2 * n - 1) + Power(0.33333, 2 * n - 1)) / (2 * n - 1)) End If End If End Function Private Function Power(ByVal b As Double, ByVal e As Integer) As Double If (e >= 0) Then If (e = 1) Then Return (b) Else Return (b * Power(b, e - 1)) End If Else If (e = -1) Then Return (1 / b) Else Return (1 / b * Power(b, e + 1)) End If End If End Function

19. Mutually Recursive Option Explicit Private Sub CMDDisplay_Click() LBLAlphabet = "" A (VSBChar.Value) End Sub Private Sub VSBChar_Change() lblChar = Chr$(VSBChar.Value) End Sub Public Sub A(c As Integer) If (Chr$(c) > "A") Then B (c) End If LBLAlphabet = LBLAlphabet + Chr$(c) End Sub Public Sub B(c As Integer) A (c - 1) End Sub

20. Fibinonacci Private Sub vsbnum_Scroll(ByVal sender As System.Object, ByVal e As System.Windows.Forms.ScrollEventArgs) Handles vsbnum.Scroll lblnum.Text = CStr(100 - vsbnum.Value) If vsbnum.Value <> 100 Then lblfib.Text = CStr(fib(100 - vsbnum.Value)) Else lblfib.Text = 0 End If End Sub Public Function fib(ByVal num As Integer) As Integer If ((num = 1) Or (num = 2)) Then fib = 1 Else fib = fib(num - 1) + fib(num - 2) End If End Function

21. Tower of Hanoi move_tower(k : INTEGER; start_peg, finish_peg, other_peg:INTEGER) is -- move tower of k disks from start_peg to finish_peg via other_peg local do if k = 1 then move_disk_0(start_peg) else move_tower(k-1, start_peg, spare_peg, finish_peg) move_other_disk(k, start_peg, finish_peg) move_tower(k-1, spare_peg, finish_peg, start_peg) end end -- move_disks

22. Code for Tower of Hanoi Public Sub movedisk(ByVal n As Integer, _ ByVal A As Integer, _ ByVal C As Integer, _ ByVal B As Integer) 'n,A,B,C; /* number to move, source pole, ‘ destination pole and spare pole respectively Dim Buffer As String If (n = 1) Then Buffer = "Move from " + CStr(A) + " to " + CStr(C) lstmove.Items.Add(Buffer) Else movedisk(n - 1, A, B, C) movedisk(1, A, C, B) movedisk(n - 1, B, C, A) End If End Sub Private Sub vsbDisks_Scroll(ByVal sender As _ System.Object, ByVal e As _ System.Windows.Forms.ScrollEventArgs) Handles _ vsbDisks.Scroll LblDisks.Text = CStr(100 - vsbDisks.Value) lstmove.Items.Clear() movedisk(100 - vsbDisks.Value, 1, 3, 2) End Sub

23. Tower of Hanoi Private Sub Form1_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load LblDisks.Text = CStr(100 - vsbDisks.Value) movedisk(100 - vsbDisks.Value, 1, 3, 2) End Sub

24. QuickSort Public Sub quicksort(ByVal first As Integer, _ ByVal last As Integer) Dim pivotpt As Integer Dim left As Integer Dim right As Integer If Not (first >= last) Then pivotpt = first left = pivotpt + 1 right = last Do Do While (left <= right) AndAlso _ (table(left) <= table(pivotpt)) left = left + 1 Loop Do While (table(right) > table(pivotpt)) right = right - 1 Loop If (right > left) Then Call swap(left, right) End If Loop While (left < right) Call swap(pivotpt, right) Call quicksort(first, right - 1) Call quicksort(right + 1, last) End If End Sub