Procedural vs functional programming

Transcription

1 2 procvsfuncprog2.nb Procedural vs functional programming Basic examples with Mathematica Hugo Touchette School of Mathematical Sciences Queen Mary, University of London Programming Seminar ate: 27 October 2010 Ideas Ë Act on lists or structures Ë Avoid loops Ë Use pure functions Ë Think global/operators/composition (vs local/sequential) Ë on't care too much about space/memory Basic example Problem: Sum integers from 1 to ü irect answer In[15]:= Sum@i, 8i, < Out[15]=

2 procvsfuncprog2.nb 3 4 procvsfuncprog2.nb In[16]:= s = 0; s = s + i, 8i, <; In[18]:= s Out[18]= I In[19]:= integerlist = Table@i, 8i, <; In[20]:= Fold@Ò1 + Ò2 &, 0, integerlist Out[20]= II In[21]:= Plus üü integerlist Out[21]= ü Time comparison test1@n_ := Module@8i<, Sum@i, 8i, n< test2@n_ := Module@8s, i<, s = 0; s = s + i, 8i, n<; s test3@n_ := Plus üü Table@i, 8i, n< iscreteplot@8timing@test1@n@@1, Timing@test2@n@@1, Timing@test3@n@@1<, 8n, 1000, , 1000<, Filling Ø None, Joined Ø True, Frame Ø True, PlotStyle Ø Thick Everything is an expression In[22]:= êê Hold êê FullForm Out[22]//FullForm= Hold@Plus@1, 2, 3, 4 In[23]:= 81, 2, 3, 4< êê FullForm Out[23]//FullForm= List@1, 2, 3, 4 In[24]:= Plus üü 81, 2, 3, 4< Out[24]= 10 In[25]:= Apply@Plus, 81, 2, 3, 4< Out[25]= 10 In[26]:= Times üü 81, 2, 3, 4< Out[26]= 24 In[27]:= Apply@Times, 81, 2, 3, 4< Out[27]= 24 In[28]:= f@x ê. f Ø g Out[28]= g@x In[29]:= MyMean@l_ := Plus üü l ê Length@l In[30]:= MyMean@81, 2, 3, 4< Out[30]= 5 2 Pure functions Ë Pure functions (Mathematica) Ë Anonymous functions Ë Lambda functions (Lambda calculus, Lisp, Python) In[31]:= Ò^2 &@c Out[31]= c 2 In[32]:= Function@u, u^2@c Out[32]= c 2 In[33]:= Ò^2 & êü 8a, b, c< Out[33]= 9a 2, b 2, c 2 = In[34]:= Map@Ò^2 &, 8a, b, c< Out[34]= 9a 2, b 2, c 2 = In[35]:= f êü Ha + b + cl Out[35]= f@a + f@b + f@c In[36]:= HÒ@@1 + Ò@@2 &L êü 88a, b<, 8c, d<< Out[36]= 8a + b, c + d< In[37]:= Nest@Ò^c &, x, 3 Out[37]= IHx c L c M c

3 procvsfuncprog2.nb 5 6 procvsfuncprog2.nb Example: Newton-Raphson iteration In[38]:= NewtonZero@f_, x0_ := FixedPoint@Ò - f@ò ê f'@ò &, x0 In[39]:= NewtonZero@BesselJ@2, Ò &, 5.0 Out[39]= ü Another example In[40]:= EmitSound@Sound@SoundNote@Ò,.1, "Percussion" & êü Range@-32, 35 Composition and pipeline ü Prefix In[41]:= füx Out[41]= f@x In[42]:= güfüx Out[42]= g@f@x ü Postfix (pipeline) In[43]:= x êê f Out[43]= f@x In[49]:= Exp êü data; êê Timing Out[49]= , Null< In[50]:= Exp@data; êê Timing Out[50]= , Null< In[51]:= ans = data; k = Length@data; While@k > 0, ans@@k = Exp@data@@k; k--; ; êê Timing Out[53]= , Null< Application: Mandelbrot set Mandelbrotset0@x_, y_ := ModuleA8z, t = 0, c = x + I y<, z = c; WhileAHAbs@z < 2.0L && Ht < 100L, z = z 2 + c; t++e; t E; ensityplot@mandelbrotset0@x, y, 8x, -1.75, 0.75<, 8y, -1.25, 1.25<, PlotPoints Ø 100, Mesh Ø False, ColorFunction Ø Hue In[44]:= x êê f êê g Out[44]= g@f@x ü Composition In[45]:= Nest@f, x, 4 Out[45]= f@f@f@f@x In[46]:= NestList@f, x, 4 Out[46]= 8x, f@x, f@f@x, f@f@f@x, f@f@f@f@x< In[47]:= Nest@1 + 1 ê Ò &, x, 3 1 Out[47]= x Listable functions In[48]:= data = TableARandomReal@, =E; Mandelbrotset2@c_, nmax_ := Length@FixedPointList@Ò^2 + c &, c, nmax, SameTest Ø HAbs@Ò2 > 2.0 &L

4 procvsfuncprog2.nb 7 8 procvsfuncprog2.nb ensityplot@mandelbrotset2@x + y I, 100, 8x, -1.75, 0.75<, 8y, -1.25, 1.25<, PlotPoints Ø 100, Mesh Ø False, ColorFunction Ø Hue Prime counting function pproc@n_ := Module@8cnt, i<, cnt = 0; If@PrimeQ@i, cnt++, 8i, 2, n<; cnt ; ListPlot@Table@8i, pproc@i<, 8i, 1, 100< ü Time comparison timingtable1 = Table@ Timing@Table@Mandelbrotset0@x, y, 8x, -1.75, 0.75, 1 ê n<, 8y, -1.25, 1.25, 1 ê n<@@1, 8n, 10, 100, 10<; timingtable2 = Table@Timing@Table@Mandelbrotset2@x + I y, 100, 8x, -1.75, 0.75, 1 ê n<, 8y, -1.25, 1.25, 1 ê n<@@1, 8n, 10, 100, 10<; ListPlot@8timingtable1, timingtable2<, Joined Ø True, Frame Ø True, PlotStyle Ø Thick List all the positive integers up to n 2. Extract the primes 3. Count what's extracted pfunc@n_ := Length@Select@Table@i, 8i, n<, PrimeQ ü Time comparison iscreteplot@8timing@pproc@n@@1, Timing@pfunc@n@@1<, 8n, 1000, , 1000<, Filling Ø None, Joined Ø True, Frame Ø True, PlotStyle Ø Thick

5 procvsfuncprog2.nb 9 10 procvsfuncprog2.nb Perfect numbers A number n is perfect if it is equal to the sum of its proper divisors (all divisors except n) or, equivalently, if the sum of all its divisors (including n) is equal to 2 n. For example, 6 is a perfect number since its proper factors, 1, 2, 3 sum to = 6. In[54]:= PerfectNumberQ@n_ := 2 n === Plus üü ivisors@n In[56]:= PerfectNumberQ@12 Out[56]= False With@8nmax = <, l = 8<; If@PerfectNumberQ@i, AppendTo@l, i, 8i, nmax<; l êê Timing , 86, 28, 496, 8128<< Time comparison With@8n = , dn = 1000<, iscreteplot@8timing@primesumproc@i@@1, Timing@primesumfunc@i@@1<, 8i, 1, n, dn<, Filling Ø None, Joined Ø True, Frame Ø True, PlotStyle Ø Thick With@8nmax = <, Select@Table@i, 8i, 1, nmax<, PerfectNumberQ êê Timing , 86, 28, 496, 8128<< Prime sums S n = 1 n n p i i=1 m Generate the list 8n, S n< n=1 primesumproc@n_ := Module@8i, ps, res<, ps = 0; res = 8<; ps = ps + Prime@i; AppendTo@res, 8i, ps ê i<, 8i, 1, n<; res primesumfunc@n_ := NestList@ 8Ò@@1 + 1, HÒ@@2 Ò@@1 + Prime@Ò@@1 + 1L ê HÒ@@1 + 1L< &, 81, 2<, n - 1

6 procvsfuncprog2.nb procvsfuncprog2.nb 1, 3, 5, 5 question Interview question: What is the combination of the numbers 1, 3, 5 and 5 with any of the arithmetic operations (+, -, *, /) that yields 16? ü Include In[1]:= Listelete@list_, elem_ := elete@list, Position@list, elem@@1, 1 In[2]:= ListOperation@l_ := Module@8res, i<, res = l@@1; res = res~l@@i~l@@i + 1, 8i, 2, Length@l, 2<; res In[3]:= numset = 81, 3, 5, 5<; In[4]:= opset = 8Plus, Subtract, Times, ivide<; In[57]:= Catch@ numset1 = Listelete@numset, n1; numset2 = Listelete@numset1, n2; opset1 = Listelete@opset, op1; res1 = n1~op1~n2; numset3 = Listelete@numset2, n3; opset2 = Listelete@opset1, op2; res2 = res1~op2~n3; n4 = numset3@@1; res3 = res2~op3~n4; If@res3 ã 16, Throw@8n1, op1, n2, op2, n3, op3, n4<, 8op3, opset2<, 8op2, opset1<, 8n3, numset2<, 8op1, opset<, 8n2, numset1<, 8n1, numset< Out[57]= 81, ivide, 5, Plus, 3, Times, 5< In[58]:= listperm = Permutations@numset Out[58]= 881, 3, 5, 5<, 81, 5, 3, 5<, 81, 5, 5, 3<, 83, 1, 5, 5<, 83, 5, 1, 5<, 83, 5, 5, 1<, 85, 1, 3, 5<, 85, 1, 5, 3<, 85, 3, 1, 5<, 85, 3, 5, 1<, 85, 5, 1, 3<, 85, 5, 3, 1<< In[59]:= listop = Flatten@Table@8i, j, k<, 8i, opset<, 8j, opset<, 8k, opset<, 2 Out[59]= 88Plus, Plus, Plus<, 8Plus, Plus, Subtract<, 8Plus, Plus, Times<, 8Plus, Plus, ivide<, 8Plus, Subtract, Plus<, 8Plus, Subtract, Subtract<, 8Plus, Subtract, Times<, 8Plus, Subtract, ivide<, 8Plus, Times, Plus<, 8Plus, Times, Subtract<, 8Plus, Times, Times<, 8Plus, Times, ivide<, 8Plus, ivide, Plus<, 8Plus, ivide, Subtract<, 8Plus, ivide, Times<, 8Plus, ivide, ivide<, 8Subtract, Plus, Plus<, 8Subtract, Plus, Subtract<, 8Subtract, Plus, Times<, 8Subtract, Plus, ivide<, 8Subtract, Subtract, Plus<, 8Subtract, Subtract, Subtract<, 8Subtract, Subtract, Times<, 8Subtract, Subtract, ivide<, 8Subtract, Times, Plus<, 8Subtract, Times, Subtract<, 8Subtract, Times, Times<, 8Subtract, Times, ivide<, 8Subtract, ivide, Plus<, 8Subtract, ivide, Subtract<, 8Subtract, ivide, Times<, 8Subtract, ivide, ivide<, 8Times, Plus, Plus<, 8Times, Plus, Subtract<, 8Times, Plus, Times<, 8Times, Plus, ivide<, 8Times, Subtract, Plus<, 8Times, Subtract, Subtract<, 8Times, Subtract, Times<, 8Times, Subtract, ivide<, 8Times, Times, Plus<, 8Times, Times, Subtract<, 8Times, Times, Times<, 8Times, Times, ivide<, 8Times, ivide, Plus<, 8Times, ivide, Subtract<, 8Times, ivide, Times<, 8Times, ivide, ivide<, 8ivide, Plus, Plus<, 8ivide, Plus, Subtract<, 8ivide, Plus, Times<, 8ivide, Plus, ivide<, 8ivide, Subtract, Plus<, 8ivide, Subtract, Subtract<, 8ivide, Subtract, Times<, 8ivide, Subtract, ivide<, 8ivide, Times, Plus<, 8ivide, Times, Subtract<, 8ivide, Times, Times<, 8ivide, Times, ivide<, 8ivide, ivide, Plus<, 8ivide, ivide, Subtract<, 8ivide, ivide, Times<, 8ivide, ivide, ivide<< In[60]:= Riffle@listperm@@1, listop@@1 Out[60]= 81, Plus, 3, Plus, 5, Plus, 5< In[61]:= allcaseslist = Flatten@ Table@ Riffle@listperm@@i, listop@@j, 8i, Length@listperm<, 8j, Length@listop<, 1 ; In[62]:= resultlist = ListOperation êü allcaseslist; In[66]:= ListPlot@resultlist, Joined Ø True, PlotRange Ø All Out[66]= In[64]:= winningpos = Position@ListOperation êü allcaseslist, 16 Out[64]= 88115<< In[65]:= Extract@allcaseslist, winningpos Out[65]= 881, ivide, 5, Plus, 3, Times, 5<< Conclusions Reform your sequential brain!

7 procvsfuncprog2.nb 13 Think lists Ë Think listable functions Ë Think global operations Ë Avoid loops Ë Avoid counters Ë Think functional programming