This Week. Trapezoidal Rule, theory. Warmup Example: Numeric Integration. Trapezoidal Rule, pseudocode. Trapezoidal Rule, implementation

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1 This Week ENGG8 Computing for Engineers Week 9 Recursion, External Application Interfacing Monday: numeric integration example, then first part of the material Wednesday 9am: rest of the new material Wednesday pm: more worked examples, see slides 3-5 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide Warmup Example: Numeric Integration Numerical integration approximates the solution to a definite integral that may not have a closed form by using a series of shapes to model the area under the curve Simplest of these is the Trapezoidal Rule, which uses thin vertical slices with a straight line at the top (forming adjacent trapeziums) a f(x) b Integral of f(x) from a to b is approximated by the sum of the areas inside the red trapeziums ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 3 Trapezoidal Rule, theory References: Wikipedia, many other web refs Animation: mations/quadrature/trapezoidal/trapezoidalaa.html After doing the maths for n equally spaced panels, the formula reduces to b a δ ( f ( x ) + f ( x ) + f ( x ) + K+ f ( x ) f ( x )) f ( x) dx n + Panel width b a δ = n 0 n Twice the middle values where xi = a + i( b a) / n = a + i *δ End-points counted once only ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 4 Trapezoidal Rule, pseudocode Trapezoidal Rule, implementation We can easily convert this to pseudocode: set sum to f (a) + f (b) set delta to the panel width For p = To add to sum Next p area = Function TrapArea(a As Double, b As Double, _ n as integer) As Double Limitations: for this quick example, we will represent the function to be integrated as a VBA function called directly from TrapArea. It is possible to specify it as a formula on the sheet, which is easier for end users (and was done in an assignment in previous years) ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 5 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 6

2 References Recursion is covered quite nicely at not quite so well in Chapra Section.4 Recursion Recursion Sometimes a solution to a problem can be expressed naturally in terms of one or more simpler versions of the same solution. For example, the factorial definition is just n! = n ( n )!, n! =, if n > 0 if n = 0 This is called a recursive definition, and can be implemented directly in most programming languages Recursion depends strongly on correctly identifying the base case, where no further subdivision occurs. ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 7 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 8 Recursive Factorial Recursion is implemented through functions or subprograms calling themselves rather than another procedure Works provided the base case is handled and there are sufficient resources to sustain the calculation Function fact(n As Integer) As Long If n = 0 Then fact = Else fact = n * fact(n-) End Function Tracing Factorial Trace the calculation as with other procedure calls fact(5) = 5 * fact(4) fact(4) = 4 * fact(3) fact(3) = 3 * fact() fact() = * fact() fact() = * fact(0) fact(0) = fact() = * = fact() = * = fact(3) = 3 * = 6 fact(4) = 4 * 6 = 4 fact(5) = 5 * 4 = 0 Each function waits for the recursive call to return Each function uses the value from the recursive call to calculate and return its factorial ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 9 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 0 Tail Recursion The Factorial function is an example of what s called tail recursion, where each call results in either no recursion (the base case), or one recursive call in an expression Tail recursion produces a sequence, called a stack of pending procedure calls Tail recursion can always be replaced by iteration, which generally consumes less resources Naughty Recursion Recursion often gets a bad name because implementing some algorithms directly produces horribly inefficient code that reevaluates things over and over Classic case is the Fibonacci sequence*: Function BadFib(n As Integer) As Long If n <= Then BadFib = Else BadFib = BadFib(n-) + BadFib(n-) End Function * Regrettably, this is the solution to Chapra s misguided exercise 6, page 47 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide

3 tree has n levels Why BadFib is Bad Tracing calls on BadFib produces a tree rather than a simple stack of calls: is evaluated three times independently F(4) F(5) F(6) F(4) Recursion in Nature Some things in nature contain elements that are smaller copies of themselves (this is called self-similarity). The branching patterns of creeks and rivers, ferns, cloud patterns and blood vessels are all examples. Anne Burns (Long Island University) has a nice paper on the web (ref below) describing general techniques for creating images such as this one. Nothing in the picture is real. ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 3 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 4 Recursive Drawings We can do a tiny bit of this with the simple drawing tools available Draw a simple figure, then several smaller copies positioned relative to the larger one Copies are drawn recursively Recursion stops when the elements get small enough that the detail can t be seen Can t (easily) convert to iteration unless only one copy (e.g., picture of A- series paper sizes) General approach: Recursive Drawing Procedure Sub DrawThing(x As Single, y As Single, size As Single) If size < MIN_SIZE Then Exit Sub draw figure at position (x, y) ' calculate position of first copy x = : y = DrawThing(x, y, size*scale_factor) ' same for other copies DrawThing(x, y, size*scale_factor) End Sub Constant describing the smallest effective size Constant describing the reduction in size for each copy Source: Wikipedia ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 5 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 6 H-Trees The following is from the Princeton reference. Recursive graphics. Simple recursive drawing schemes can lead to pictures that are remarkably intricate. For example, an H-tree of order n is defined as follows: The base case is do nothing for n = 0. The reduction step is to draw, within the unit square three lines in the shape of the letter H [then] four H-trees of order n, one connected to each tip of the H with the additional provisos that the H-trees of order n are centered in the four quadrants of the square, halved in size. Question: What size will the entire picture be compared to the first H? (This is related to Zeno s Paradox about Achilles and the Tortoise) H-Trees Apart from drawing a canvas, the algorithm for drawing an H-Tree centred on (x,y) is just (x+hsize,y HTree(x,y,size) :- If size < MIN_SIZE Then Exit Sub (x,y) hsize = size/ ' half-size (x+hsize,y-hsize) hsize) Draw horizontal line through (x,y) of length size Draw vertical lines through (x-hsize,y) and (x+hsize,y) HTree x-hsize, y+hsize, hsize ' bottom left HTree x-hsize, y-hsize, hsize ' top left HTree x+hsize, y-hsize, hsize ' top right HTree x+hsize, y+hsize, hsize ' bottom right Use a separate subprogram to draw the lines, so you can apply the attributes consistently and once only size hsize ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 7 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 8 3

4 H-Tree Demo See the demo workbook. To make it more useful as a learning tool, the following have been added a ClearDrawing subroutine a DrawCanvas subroutine (shape reference saved) a depth parameter, indicating how many calls are active colours, derived from the depth a visible stack, showing the depth in boxes timings, to demonstrate inefficiencies in VBA s collections Most important parts are the subprograms DrawHTree, HTree itself and DrawLine Office Interfacing All MS Office applications can start and interact with any other Office app For example, Excel can start Word and transfer data from a sheet to a table Word can start Outlook and send s (if an server is available) Excel can obtain data from an Access database (though this requires a lot of fiddling around) Depending on your processor, the picture will take a couple of minutes to draw, and the drawing rate will slow down as each line is added to the shape collection. ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 9 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 0 Example: Creating a Word Doc Object Reference Settings ' Example adapted from Shepherd, Excel VBA Sub Test_Word() Dim objwordapp As Word.Application ' reference to Word (the app) Dim objworddoc As Word.Document ' ref to a newly created doc Set objwordapp = CreateObject("Word.Application") Set objworddoc = objwordapp.documents.add With objworddoc.sections().range.text = "My new Word Document".SaveAs "c:\mytest.doc" ' must use full pathname.close End With ' These are required to make sure MS Word shuts down completely Set objworddoc = Nothing Set objwordapp = Nothing End Sub Unfortunately this fails to satisfy the compiler. The error points to Word.Application Select Tools References on the VBE menu, scroll to Microsoft Word * Object Library and check. Press OK. * or largest value: 0=Office00, =Office003, =Office007 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide This is the end of the new VBA material (all together now, a big groan of disappointment) Some short worked examples of the kind that could be set for the final exam will follow. A bearing is an angle measured clockwise from North Write a VBA subprogram that candy-stripes the active worksheet. Candy-striping fills the background of every second row with a faint grey colour. Stop with the first empty cell in column A. Use the macro recorder to gather information about colouring rows, then work on the loop to apply the fills. A back bearing is the bearing that is exactly opposite a given bearing. Bearings lie between 0 and 360 degrees. For example, if I'm sighting a landmark at that has a compass bearing of 0 degrees (slightly South of East), the back bearing is 90 degrees (slightly North of West). The back bearing is the bearing from the landmark back to me. Write a VBA function BackBearing that returns the back bearing for any given bearing, expressed in degrees between 0.0 (inclusive) and (exclusive). ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 3 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 4 4

5 3 A worksheet contains rainfall data for Liverpool, NSW for the month of April 009, downloaded from A domestic water tank is fed from the guttering of a house. Given the roof area, tank capacity and initial amount of water in the tank, complete the worksheet using a VBA procedure. Assume no water is drawn from the tank over the month. Analysis: how much water does mm of rain produce per m of area? Named cells RoofArea and TankCapacity Summary Recursion is a natural way to express some algorithms tail recursion can be replaced by iteration care must be taken to avoid recursion that re-evaluates the same thing over and over drawing recursive pictures may help in understanding how recursion works Sometimes you need to think creatively about solutions Office apps can create and manipulate documents and data through the use each other s object models expressed in VBA. VBA fills in this column ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 5 ENGG8 UNSW, CRICOS Provider No: 00098G W9 slide 6 5