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XNÉIO 0 1 ° XNÉCT 0 1E²13 ° XCÉFC 1 6 .,*0_² ° XNÉRL 0 530511967 ° XCÉPR 1 1 ° XCÉLX 1 5 TEACH ° *(1996 4 6 16 11 29 328) ° FBIND ÉFX 'BIND' 'Σ Binding rules of APL2' ° ''' 1. Brackets bind to the left''' ° ''' 2. Left arrow binds to the left''' ° ''' 3. Dyadic operator binds to the right''' ° ''' 4. Strand notation binding''' ° ''' 5. Operator binds to its left''' ° ''' 6. Function binds to its left''' ° ''' 7. Function binds to its right''' ° X ''' 8. Left arrow binds to its right'',r' ° *(1996 4 6 16 11 56 436) ° XFDEGREES ÉFX 'U╜DEGREES W' 'U╜180⌡W÷Ω1' ° *(1997 9 14 12 30 44 504) ° XFDISCLAIMER ÉFX 'DISCLAIMER' 'Σ Copyright statement' 'disclaimer' ° *(1996 4 6 16 11 56 436) ° FDISPLAY ÉFX 'D╜S DISPLAY A;ÉIO;R;C;HL;HC;HT;HB;VL;VB;V;W;N;B' ° 'Σ DISPLAY A GENERAL ARRAY IN PICTORIAL FORM' ° 'Σ NORMAL CALL IS MONADIC. DYADIC CALL USED ONLY IN' ° 'Σ RECURSION TO SPECIFY DISPLAY RANK, SHAPE, AND DEPTH.' 'ÉIO╜0' ° '»(0=ÉNC ''S'')/''S╜µA''' 'R╜╞µ,S Σ PSEUDO RANK.' ° 'C╜''┐┌└┘'' Σ UR, UL, LL, AND LR CORNERS.' ° 'HL╜''─'' Σ HORIZONTAL LINE.' ° 'HC╜HL,''Θ╕'',HL,''~+ε'' Σ HORIZONTAL BORDERS.' ° 'HT╜HC[(0<R)⌡1+0<╞²1╞,S]' 'ΣW╜,0╧■╞0µΓ(1⌐µA)╞A' ° 'HB╜HC[3+3╛(''2⌡~A╧«A'' ÉEA ''1+╞ε0⌡(1⌐⌡/µA)╞,A'')+3⌡1<µµS]' ° 'VL╜''│'' Σ VERTICAL LINE.' ° 'VB╜VL,''Φ╟'' Σ VERTICAL BORDER.' ° 'V╜VB[(1<R)⌡1+0<²1╞²1╟,S]' ° '»(0εµA)/''A╜(1⌐µA)µΓ╞A'' Σ SHOW PROTOTYPE OF EMPTIES.' ° '╕(1<╧A)/GEN' '╕(2<µµA)/D3' ° 'D╜«A Σ SIMPLE ARRAYS.' 'W╜1╞µD╜(²2╞1 1,µD)µD' ° 'N╜²1+1╟µD' '╕(0=µµA)/SS' ° 'D╜(C[1],V,((W-1)µVL),C[2]),((HT,NµHL),[0]D,[0]HB,NµHL),C[0],(WµVL),C[ ° 3]' '╕0' 'SS:HB╜((0 '' '')=╞0µΓA)/'' -''' ° 'D╜'' '',('' '',[0]D,[0]HB,Nµ'' ''),'' ''' '╕0' ° 'GEN:D╜«DISPLAY■A Σ ENCLOSED ...' 'N╜Dδ.⌠'' ''' ° 'D╜(Nδ~1ΦN)≡D' 'D╜(δ≡~'' ''╤D)/D' 'D╜((1,µS)µS)DISPLAY D' ° '╕(2≥µ,S)╟D3E,0' 'D3:D╜0 ²1╟0 1╟«ΓA Σ MULT-DIMENSIONAL ...' ° 'W╜1╞µD' 'N╜²1+1╟µD' ° 'D╜(C[1],V,((W-1)µVL),C[2]),((HT,NµHL),[0]D,[0]HB,NµHL),C[0],(WµVL),C[ ° 3]' 'D3E:N╜²2+µ,S' ° X 'V╜C[Nµ1],[0]VB[1+0<²2╟,S],[0](((²3+╞µD),N)µVL),[0]C[Nµ2]' 'D╜V,D' ° *(1996 4 6 16 11 56 436) ° FEXIT ÉFX 'EXIT' 'Σ Exit from function' '''To log off type: )OFF''' ° X '╕' ° *(1996 4 6 16 11 56 436) ° FGO ÉFX 'GO;T;E;B' 'Σ Expression driver' 'L0:B╜E╜''''' 'æ╜'' ''' ° 'T╜æ' '╕(^/'' ''=T)/L0' '╕((^/'')OFF ''=5╞6╟T)doif ''EXIT'')/0' ° '╕(('':''εT)doif ''B╜evaldd (+/^\'''' ''''=T)╟T'')/L0' ° '''E╜ÉEM'' ÉEA T' '╕(0=µ,E)/L0' '╕B/L0' ° '''This is not a valid APL2 expression''' 'æ╜''*''' ° X '╕(''?''⌠╞1╟æ)/L0' 'E' '╕L0' ° *(1996 4 6 16 11 56 436) ° FGRAD ÉFX 'U╜GRAD A0;AI;SS0;SS1' ° 'Σ Gradient method of function minimization (R and X are global)' ° 'SS0╜R SS A0 FUN X' 'AI╜3 3µA0' 'U╜SS0-R SS(AI+0.01⌡AI⌡ID 3)FUN X' ° X 'U╜A0 MIN U÷A0' ° *(1996 4 6 16 11 56 436) ° FGRADIENT ÉFX 'GRADIENT;D' 'Σ Explanation of gradient minimization' ° '''The basic idea is to find a direction of search, such that the''' ° '''value of SS will decrease the fastest''' '0 2 showfn ''GRAD''' ° '''In line [2] of function GRAD, we first compute the sum of squares'' ° ' ° '''associated with A0. Next we want to find the rate of change (slope) ° ''' ° '''of function SS(A) in each of the components (directions) of A0. To' ° '' '''do that we prepare a square array of vectors A0''' ° '0 3 showfn ''GRAD''' 'do' ° '''The array AI+0.2⌡AI⌡ID 3 adds to each vector of AI a small incremen ° t''' ° '''in each of its components. Finding the difference between SS0 and t ° he''' ° '''sum of squares of each of these incremental vectors gives the rate ° of''' '''change in each of the component directions''' ° '0 4 showfn ''GRAD''' ° '''Finally to get the direction of descent, we divide vector U by A0.' ° '' '''This vector is then used as right argument to function MIN''' ° '0 5 showfn ''GRAD''' 'do' '''MINIMIZING SS GIVEN A DIRECTION'',r' ° '''By its nature a sum of squares of deviations from an mean value''' ° '''would be expected to behave in a way similar to the function Y=X*2' ° '' ° '''which is a parabola. The estimate of minimum value of function SS(A ° )''' ° '''can be made based on that assumption. That is the objective of''' ° '''function MIN.''' 'do' '1 showfn ''MIN''' ° '''The left argument to MIN is the current value of the constant, and' ° '' ° '''the right argument is the direction of search. Let us take a closer ° ''' '''look at the algorithm of MIN''' 'do' '0 2 showfn ''MIN''' ° '''Since SS0 is required here, it is calculated, but only if not known ° ''' '''(GRAD may not be the only place where MIN will be applied.)''' ° '0 3 showfn ''MIN''' ° '''Since we cannot assume that SS(V+W) will be smaller than SS(V),''' ° '''the variable T (in units of W) is initialized to ²1 and +1''' 'do' ° '0 4 showfn ''MIN''' ° '''To insure that only small increments of W (relative to V) are appli ° ed,''' ° '''the length of W is normalized to be 1÷100 of the length of V'',r' ° '0 5 showfn ''MIN''' ° '''Line [5] calculates SS for offsets at ²1 0 and +1''' 'do' ° '''Before we look at the remaining code, let us review some simple''' ° '''algebra. We postulate a quadratic function of the form'',r' ° ''' Y ╜ a + bX + cXX'',r' ° '''The minimum (or maximum) can be found by taking derivative and by'' ° ' '''setting Y'''' to zero'',r' 'do' ° ''' 0= b + 2cX(min) or X(min) = -b÷2c'',r' ° '''Since we know this function for X= ²1 0 1, the problem of finding'' ° ' '''X(min) is reduced to the solution of two equations'',r' ° ''' 1╙SS2 ╜ 2╙SS2 - b + c'',r' ° ''' 3╙SS2 ╜ 2╙SS2 + b + c'',r' ° '''From these we find that the value of X(min) is given by'',r' ° ''' (-/SS2[1 3])÷D╜(+/SS2[1 3])-2⌡SS2[2]''' 'do' ° '''If D>0 then we are dealing with a maximum. In that case we merely'' ° ' ° '''set X(min) to an offset. Lines [6 7] of MIN deal with these cases'' ° ,r' '0 6 7 showfn ''MIN''' 'do' '0 8 showfn ''MIN''' ° '''Since we cannot be sure that SS(A) will actually behave as a parabo ° la''' ° '''we merely add the value of X(min) for which SS(A) should be smalles ° t''' '''to T, and compute the associated value of SS'',r' ° '0 9 showfn ''MIN''' ° '''Finally we select the constants associated with the smallest value' ° X'' '''of SS1''' 'do' 'endd' ° *(1997 8 21 13 29 57 580) ° FHELP ÉFX 'HELP;N;I;T' 'Σ Help to student' '''WHAT TO DO'',r' ° ''' ° To get out of the lesson''' ''' ENTER: EXIT''' ° ''' ° To log off APL2''' ° ''' FIRST, ENTER: EXIT THEN ENTER: )OFF''' ° ''' ° To get help''' ''' ENTER: HELP''' ° ''' ° When you see the prompt on a blank line''' ° ''' ENTER AN APL2 EXPRESSION - OR JUST PRESS: ENTER''' 'do' ° ''' ° If you get this line'',r' ° '''This is not a valid APL2 expression'',r,''*'',r' ° ''' YOU CAN EITHER''' ''' A. PRESS: ENTER''' ° ''' B. PRESS: ? and then ENTER to see what was incorrect''' ° ''' in your expression causing that response'',r' 'do' ° '''HINTS'',r' ° '''This lesson is made up of '',(«N╜5),'' components named TEACHx, whe ° re''' '''the x stands for a digit:'',r' 'I╜0' ° 'L0:T╜''TEACH'',(«I╜I+1)' 'T,'' '',1╟notb(ÉCR T)[2;]' '╕(N>I)/L0' ° 'do' ° '''You may review either of these components separately. To do that,'' ° ' ° '''first enter EXIT (and RETURN), then enter the name of the lesson.'' ° ' '''component (e.g. TEACH4).'',r' ° '''To re-start the lesson, just enter TEACH'',r' ° '''When the screen fills up, it is a good idea to move the cursor to'' ° ' '''the start of a convenient paragraph, and press ENTER'',r' ° '''You may also wish to press PAGE UP to review the prevous pages.''' ° 'do' '''RESOURCES'',r' ° '''You may also enter ONE of the following words at a time'',r' ° '''BIND List the binding rules''' ° '''GRADIENT To discuss function minimization using a gradient''' ° X 'endd' ° *(1996 4 6 16 11 56 436) ° XFID ÉFX 'u╜ID w' 'ΣDD' 'u╜(2µw)µ1,wµ0' ° *(1996 4 6 16 11 56 436) ° FMIN ÉFX 'U╜V MIN W;SS1;T' ° 'Σ Find minimum given direction W and constants V (global R, X and SS0 ° )' 'T╜(0=ÉNC ''SS0'')doif ''SS0╜R SS V FUN X''' 'T╜²1 0 1' ° 'W╜0.01⌡W⌡((+/V*2)÷+/W*2)*0.5' ° 'SS1╜²1ΦSS0,R SS((2 3µV)+1 ²1°.⌡W)FUN X' ° '╕((0≥D╜+/SS1[1 3]-2⌡SS1[2])doif ''T╜T,2⌡╞(SS1=╛/SS1)/²1 0 1'')/L0' ° 'T╜T,(-/SS1[1 3])÷D' 'L0:SS1╜SS1,R SS(V+T[4])FUN X' ° X 'U╜V+W⌡(SS1=╛/SS1)/T' ° *(1996 4 6 16 11 56 436) ° XFRADIANS ÉFX 'U╜RADIANS W' 'U╜ΩW÷180' ° *(1997 7 24 13 6 36 412) ° FTEACH ÉFX 'TEACH' 'Σ Start lesson #22: APL2 by Zdenek V JIZBA' ° 'exit ''TEACH''' 'initialize' 'TEACH1 Σ Matrix divide' ° 'TEACH2 Σ The grocery problem revisited' ° 'TEACH3 Σ Fitting observed data to a polynomial' ° 'TEACH4 Σ Fitting obseved data to other equations' ° X 'TEACH5 Σ Function minimization' 'problems' ° *(1997 7 23 13 48 46 616) ° FTEACH1 ÉFX 'TEACH1;ROWS;COLS;BILL;BUY;PRICE;COST;INB;GRNO' ° 'Σ Solving linear equations with dyadic domino' ° ''' MATRIX DIVIDE'',r' 'BUY╜3 3µ2 1 1 1 2 2 3 2 1' ° 'PRICE╜0.3 0.4 0.5' 'COST╜BUY+.⌡PRICE' ° '''In the previous lesson we learned about matrices, geometrical vecto ° rs''' ° '''(as contrasted with APL2 vectors, -- a more general concept), and w ° e''' ° '''studied the APL2 primitive function called MATRIX INVERSE.'',r' ° '''As an example of the use of matrix inverse we solved the grocery''' ° '''price of fruit problem: Given the purchase of fruit by three''' ° '''customers, find the unit cost of each fruit. We defined a matrix''' ° '''BUY and two vectors COST and PRICE'',r' ° 'show ''COST BUY PRICE''' ° '''In this problem, PRICE is the unknown'',r' ° 'ROWS╜πΓ■''CUSTOMER 1'' ''CUSTOMER 2'' ''CUSTOMER 3''' ° 'COLS╜'''' ''APPLES'' ''ORANGES'' ''BANANAS'' '' COST''' ° 'BILL╜3 1µΓ[2]3 6µ(ÉAV[33 37],''0.59'')«COST' 'COLS,[1]ROWS,BUY,BILL' ° 'do' ° '''To solve this problem, we first found the inverse of matrix BUY, th ° en''' '''we multiplied this inverse by COST'',r' ° 'show ''INB╜ÆBUY'' ''INB+.⌡COST''' ° '''In the dyadic version of the DOMINO primitive, this procedure can b ° e''' '''accomplished in a single step'',r' 'show ''COSTÆBUY''' ° '''This dyadic function is called MATRIX DIVIDE, since it is a''' ° '''generalization of division into the matrix domain. The right argume ° nt''' ° '''to matrix divide should be a matrix. The left argument can be a vec ° tor''' ° '''or a matrix. Suppose that in the grocery problem, we compare the co ° sts''' '''of the same items from two or more groceries'',r' ° 'show ''É╜GRNO╜3 3µ1.5 2.1 2.2 1.3 1.85 1.9 1.5 2.1 2.25'' ''(φGRNO)ÆB ° UY''' ° '''The columns of the result return the unit costs of each item.'',r' ° 'BILL╜(φ3 3µ1.5 2.1 2.2 1.3 1.85 1.9 1.5 2.1 2.25)ÆBUY' ° '('''' ''STORE 1'' ''STORE 2'' ''STORE 3''),[1](3 1µ1╟COLS),BILL' ° 'do' ° '''Although matrix BUY is a square one, this is not a requirement, as' ° '' ° '''we will see shortly. For matrix divide to work, however there are'' ° ' '''some limitations:'',r' ''' A. Right argument'',r' ° ''' 1. If it is a vector, there is no matrix divide. The result ° is''' ° ''' related to the monadic version and the following identity ° ''' ''' holds:'',r' ° ''' V1+.⌡ÆV2 ╜ ╕ V1ÆV2 or''' ° ''' (φM)+.⌡ÆV ╜ ╕ MÆV2'',r' ° ''' 2. If the right argument is a matrix, the number of columns' ° '' ° ''' must not exceed the number of rows. Also no column may be ° ''' ° ''' the same as the matrix product of the remaining columns.' ° '' ''' This requirement is called LINEAR INDEPENDENCE.''' ° 'do' ''' B. Left argument'',r' ° ''' 1. If it is a vector, its length must match the number of''' ° ''' rows in the right argument.'',r' ° ''' 2. If the left argument is an array, the number of rows''' ° ''' must be the same as the number of rows of the right''' ° ''' argument.'',r' ° ''' C. If both arguments are simple scalars, Æ is the same as ÷''' ° 'do' ° '''Next let us describe the meaning of XÆY when Y is not a square''' ° X '''matrix.''' 'endd' ° *(1997 7 23 13 54 59 692) ° FTEACH2 ÉFX 'TEACH2;D;BUY;PRICE;COST' 'Σ Least squares' 'D╜ÉAV[37]' ° ''' FITTING BY LEAST SQUARES'',r' ° 'BUY╜4 3µ2 1 1 1 2 2 3 2 1 2 2 2' 'PRICE╜0.3 0.4 0.5' ° 'COST╜BUY+.⌡PRICE' ° '''Suppose that we add a fourth customer to our list. This one bought' ° '' '''two apples, two oranges, and two bananas, paying $2.40'',r' ° 'show ''BUY'' ''COST''' ° '''We can still use the same expression to determine the PRICE of each ° ''' '''unit'',r' 'show ''COSTÆBUY''' ° '''Adding an extra customer really did not do anything. The new data'' ° ' ° '''is redundant. But suppose that the price of the fruit is not on a'' ° ' '''unit basis, but rather on a per/pound basis. If the fourth''' ° '''customer selected somewhat smaller than average fruit, the cost''' ° '''may have been $2.35 instead of $2.40. Will the same expression''' ° '''work, and if yes, what does it mean?'',r' ° 'show ''COST[4]╜2.35'' ''8 2«.005+COSTÆBUY''' ° '''(Why did we insert the expression ''''8 2«.005+''''?)''' 'do' ° '''The numbers we now get represent the AVERAGE cost of each''' ° '''fruit. To explain precisely the underlying mathematics of the''' ° '''calculation would take us too far into a subject called''' ° '''MATHEMATICAL STATISTICS. We can however explain the basic idea.''' ° 'do' ° '''Let us assume that the grocery store would like to receive on the'' ° ' ° '''average the amounts of 0.30 0.40 and 0.50 each for one apple, one'' ° ' ° '''orange and one banana. To do that, they find the average weight of' ° '' ° '''these fruit, and price the per/pound accordingly. When we select''' ° '''specific fruit, we find that the deviation from the average can be' ° '' '''positive or negative'',r' 'show ''PRICE-COSTÆBUY''' ° '''We would like to treat the negative numbers the same as the positiv ° e''' ° '''numbers. There are several ways to do that, but as Karl Gauss showe ° d''' ° '''in 1795, the best way is to square these differences and add them u ° p'',r' 'show ''+/(PRICE-COSTÆBUY)*2''' ° '''This quantity, called the SUM OF SUARES (often abbreviated SS)''' ° '''should be minimized for a best estimate. This is indeed what''' ° '''the expression COSTÆBUY actually does!''' 'do' ° '''The method of LEAST SQUARES is a very general one. It is the''' ° '''foundation of a great deal of applied statistics. We will not''' ° '''deal with this extensive subject here, except to point out that''' ° '''APL2 is an ideal language for solving problems of a statistical''' ° '''nature. We will give more examples including some techniques for''' ° X '''the solution of very difficult problems.''' 'endd' ° *(1997 7 24 12 47 9 464) ° FTEACH3 ÉFX 'TEACH3;A;X;Y;A1;YOBS1;YOBS2;SS1N;SS1R;YTH' ° 'Σ Fitting curves' '''FITTING CURVES'',r' ° '''Let us suppose that a scientist has reason to believe that every''' ° '''time a value is set for variable X, the value for variable Y''' ° '''should be determined by a polynomial of the form'',r' ° ''' Y = A + (B⌡X) + (C⌡XX)'',r' ° '''(As before we write XX to represent X*2)'',r' ° '''Furthermore, let us suppose that unknown to the scientist, the''' ° '''value of constants A, B and C is 4 2 ²0.3''' 'do' ° '''The polynomial equation can be written as an APL2 expression'',r' ° ''' A[1]+X⌡A[2]+X⌡A[3] where'',r' ° '''Why is the expression "a+X⌡b+X⌡c" equivalent to "a+(bX)+(cXX)"?'',r ° ' 'show ''A╜4 2 ².3''' ° '''The belief that there is a specific relation between X and Y is cal ° led''' ° '''a HYPOTHESIS. To demonstrate the validity of the hypothesis, the''' ° '''scientist must do the following:'',r' ° ''' 1. Take a number of measurements of X and Y''' ° ''' 2. Find the "best" constants A, B and C''' ° ''' 3. Show that the polynomial is "a good fit".''' 'do' ° '''Since the values of X can be selected before an experiment, let us' ° '' ° '''assume that this choice is without error. (If we cannot select X''' ° '''without error, the theory gets much more complex.)'',r' ° 'show ''X╜∞10''' ° '''We can now calculate the values of Y that would be predicted by''' ° '''theory given vector A'',r' 'show ''Y╜A[1]+X⌡A[2]+X⌡A[3]'' ''Y''' ° 'do' ° '''We can verify that these values are correct, by recovering A from'' ° ' '''X, Y and the equation'',r' 'show ''YÆ1,X°.*1 2''' ° '''In a typical experiment, the scientist obtains Y by making measurem ° ents.''' ° '''We can simulate this process by adding a random error to the values ° ''' ° '''of vector Y. We will call the vector of observed Y''''s YOBS1'',r' ° 'show ''YOBS1╜Y+.01⌡²10+?10µ19'' ''YOBS1''' ° '''The results can be used to estimate vector A. We will call this vec ° tor''' '''A1 because it is only an ESTIMATE of A'',r' ° 'show ''É╜A1╜YOBS1Æ1,X°.*1 2''' ° '''If the errors of measurement are greater, our estimate of A will''' ° '''deviate more from the theoretical values'',r' ° 'show ''YOBS2╜Y+.05⌡²10+?10µ19'' ''YOBS2''' ° 'show ''YOBS2Æ1,X°.*1 2''' ° '''At this point we have taken two of the three steps: We have made''' ° '''the observations, and we have estimated the PARAMETERS (namely the' ° '' ° '''vector A). We still need to determine to what extent the hypothesis ° ''' '''is correct. Here is a brief outline of that process:''' 'do' ° '''The basic idea is to show that the hypothesis explains the outcome' ° '' ° '''of the experiment better than no hypothesis, (also called the NULL' ° '' ° '''HYPOTHESIS). If we have no theory, then the SUM OF SQUARES of the'' ° ' ° '''observations is defined as the SUM OF DEVIATIONS FROM THE AVERAGE.' ° '' '''We will call this quantity SS1N'',r' ° 'show ''SS1N╜+/(YOBS1-(+/YOBS1)÷µYOBS1)*2''' ° '''On the other hand given the estimated values of vector A, we can''' ° '''also obtain the theoretical values of Y'',r' ° 'show ''YTH╜(1,X°.*1 2)+.⌡A1''' ° '''and from these obtain the RESIDUAL SUM OF SQUARES'',r' ° 'show ''SS1R╜+/(YOBS1-YTH)*2''' ° '''Comparing the values of SS1N and SS1R (and applying a number of''' ° '''special equations), it is possible to determine the LIKELIHOOD''' ° '''that the hypothesis is correct. For YOBS1 there is not much doubt'' ° ' '''that the polynomial makes a good prediction'',r' ° 'show ''SS1N'' ''SS1R''' ° '''(You might also try these calculations for the estimate YOBS2)''' ° X 'endd' ° *(1997 7 24 12 53 46 636) ° FTEACH4 ÉFX 'TEACH4;A;G;R;R1' 'Σ Fitting trigonomrtric functions' ° ''' FITTING TRIGONOMETRIC FUNCTIONS'',r' ° '''Suppose now that instead of a polynomial, the equation that the''' ° '''scientist desires to fit involves trigonometric functions'',r' ° ''' R = a + b(cosα) + c(cos2α)'',r' ° '''Here again we will set vector A to contain the values a,b and c, an ° d''' '''set'',r' 'show ''A╜1 2 3''' ° '''Then this equation can represented by the following expression'',r' ° ''' R╜A[1]+(2ΩG°.⌡1 2)+.⌡A[2 3]'',r' ° '''where G is the vector of angles expressed in radians.''' 'do' ° '''In the theory of least squares, this is treated exactly the same''' ° '''as the problem of fruit in the grocery store. We merely treat the'' ° ' ° '''cosines as unknown variables. You may think of them as if we made'' ° ' '''the following transformations'',r' ° ''' X╜cosα and Y╜cos2α.'',r' '''Given angles G'',r' ° 'show ''G╜RADIANS 0,10⌡∞9''' '''We find the values for R'',r' ° 'show ''É╜R╜A[1]+(2ΩG°.⌡1 2)+.⌡A[2 3]''' ° '''and given observations R1'',r' 'show ''R1╜R+.001⌡(?10µ19)-10''' ° '''we get an estimate of A, say A1'',r' 'show ''R1Æ1,2ΩG°.⌡1 2''' ° '''A more difficult problem arises if the theory leads us to the''' ° '''following equation'',r' ''' R ╜ a + cos(b+α) - cos(c+2α)'',r' ° '''The theory of least squares can still be applied, but now the''' ° '''equations no longer fall in the domain of linear algebra. The''' ° '''problem is still soluble but requires a different strategy.''' ° '''Often, tailor made algorithms have to be optimised for each''' ° X '''individual problem.''' 'endd' ° *(1997 7 24 13 2 10 292) ° FTEACH5 ÉFX 'TEACH5;A;A0;A1;D;R;X;FN;FUN;SS;LOOP' ° 'Σ Function minimization' ''' FUNCTION MINIMIZATION'',r' ° '''There are general techniques for the solution of this type of''' ° '''problem, but these may not be efficient, and in many cases not''' ° '''even reach the true least squares solution. Just to illustrate''' ° '''how these techniques work, we will attempt to find the constants''' ° '''a, b and c in the following equation.'',r' ° ''' R ╜ cos(a+x) + cos(b+2x) + cos(c+3x)'',r' ° '''We will use the method of gradients, also known as the method of''' ° '''STEEPEST DESCENT. We begin by defining vector A to contain the''' ° '''values of a, b and c'',r' 'show ''A╜1 2 3''' ° '''This time when translating the equation into an APL2 expression,''' ° '''we define a function that calculates R, given A and X. We also must ° ''' ° '''define the function in such a way that it will calculate R for more ° ''' '''than one set of A''''s.'',r' ° 'show ''FN╜''''R╜A FUN X'''' ''''Σ cos(1╙A+X)+cos(2╙A+2⌡X)+cos(3╙A+3⌡X ° )''''''' ° 'show ''FN╜FN,Γ''''╕((2=µµA)doif ''''''''R╜φπ(Γ[2]A) FUN■ ΓX'''''''')/ ° 0''''''' 'show ''ÉFX FN,Γ''''R╜+/2ΩA+[2]X°.⌡∞3''''''' ° '1 showfn ''FUN''' 'do' ° '''Next, for convenience, we also define a function to calculate the'' ° ' ° '''sum of squares with α╜R and ∙╜A FUN X where A may be an array'',r' ° '1 show ''SS:+≡((φ(Φµ∙)µα)-∙)*2''' ° '''To set up the problem, we define X, and observations R'',r' ° 'show ''X╜RADIANS 0,10⌡∞9''' 'show ''R╜(A FUN X)+.001⌡(?10µ19)-10''' ° '''The basic idea for the general solution is to start with an initial ° ''' ° '''estimate of A, find the associated SS, and then try to migrate A''' ° '''such that SS is constantly reduced. The expectation is that with''' ° '''enough iterations, the values of A will approach the least squares' ° '' '''solution.''' 'do' ° '''Let us begin with an initial value of A0'',r' ° 'show ''A0╜1.5 1.5 2.5'' ''A-A0''' ° '''Function GRAD takes as the right argument this estimate, and return ° s''' '''the new value of A with reduced value of SS'',r' ° '1 showfn ''GRAD''' ° '''For more details on the gradient approach, type GRADIENT'',r' ° 'show ''R SS A0 FUN X'' ''A1╜GRAD A0'' ''R SS A1 FUN X''' ° '''One iteration seldom reaches the minimum. We can define function''' ° '''LOOP to iterate more than once, and see the rate of convergence''' ° '''with the gradient technique'',r' ° 'show ''FN╜''''LOOP W;S;SS0'''' ''''Σ minimization with gradient'''' ' ° '''S╜0''''''' ° 'show ''FN╜FN,''''L0:A1╜GRAD A1'''' ''''²1╞S╜S,SS0╜R SS A1 FUN X'''''' ° ' 'show ''ÉFX FN╜FN,''''╕(=/²2╞S)/0'''' ''''╕(0<W╜W-1)/L0''''''' ° '1 showfn ''LOOP''' 'do' ° '''Note that exit from LOOP ocurrs if the new value of S is unchanged' ° ',r' 'show ''LOOP 10''' ° '''The result will not necessarily converge to the least squares''' ° '''solution, because it may get caught in what is known as a LOCAL''' ° '''MINIMUM. There are ways of avoiding such local minima. We can''' ° '''check whether the estimate A0 is approaching the true value of A'', ° r' 'show ''A'' ''A0'' ''A-A0'' ''A-A1''' ° '''In this example, we may not be in a local minimum, but in a valley. ° ''' ° '''Think of a river gorge. The steepest slopes point down to the''' ° '''river, but the direction of global minimum is at right angles to''' ° '''this slope. We can ammend LOOP by adding an extra search in the''' ° '''direction A1-A0'',r' ° 'show ''ÉFX (²1╟FN),''''A1╜A MIN A1-A0'''' ''''²1╞S,R SS A1 FUN X'''', ° ²1╞FN''' '1 showfn ''LOOP''' 'do' '''And try again'',r' ° 'show ''A1╜A0'' ''LOOP 10''' '''This time the search converges.''' ° X 'show ''A1'' ''A-A1''' 'endd' ° *(1997 7 13 12 28 49 504) ° Faddquote ÉFX 'u╜addquote w' ° 'Σ Put quotes around a string, and double existing quotes' ° X 'u╜ÉAV[40],((1+w=ÉAV[40])/w),ÉAV[40]' ° *(1996 4 6 16 11 56 436) ° XFaq ÉFX 'U╜aq W' 'U╜Γaddquote W' ° *(1997 7 24 13 20 38 476) ° Fav ÉFX 'av;A;N;I;ÉIO' 'Σ Display characters in ÉAV' 'ÉIO╜0' ° 'A╜22 78µ'' ''' 'N╜3 0«φ12 22µ1+∞356' 'A[;,(6⌡∞12)°.+2 3 4]╜N' ° 'A[;6+6⌡∞12]╜φ12 22µÉAV' 'ΣA[8 10 13;6]╜'' ''' 'A[13;6]╜'' ''' ° X 'A[14+∞8;68 69 70 72]╜'' ''' 'A' ° *(1991 11 11 8 25 13 316) ° Fdate ÉFX 'u╜date w' 'Σ Format date and time of day' 'u╜«6╞w' ° X 'u╜('' ''⌠u)Γu' 'u╜εu,■''-- .. ''' ° XCdig 1 10 1234567890 ° *(1997 9 9 13 0 45 372) ° Fdisclaimer ÉFX 'disclaimer' 'Σ Copyright statement' ° '(10µ'' ''),''Copyright, Z. V. Jizba, 1995,1996,1997'',r' ° ''' This and subsequent workspaces labelled TEACHxx are made available ° ''' ° '''at no cost to anyone who desires to learn how to use effectively''' ° '''the IBM/OS2 version of APL2.'',r' ° '''This software is provided "AS IS" with no WARRANTY of any kind, eit ° her''' ° '''express or implied. Any risk in its use resides with you, the user ° of''' '''these tutorials.'',r' ''' ACKNOWLEDGEMENTS'',r' ° ''' In writing these tutorials, I am greatly indebted to Roy Sykes, wh ° ose''' ° '''excellent lectures increased my understanding of the language.''' ° '''Discussions with the late Harry Bertucelli clarified a number of''' ° '''concepts and caused me to change some improper terminology that was ° ''' ° '''used in previous versions of these tutorials. Mr. Benjamin Archer'' ° ' ° '''kindly checked out a nearly final version, bringing to my attention ° ''' '''some ommisions, misspellings, and invalid terminology.'',r' ° X '''(PRESS ENTER to continue)''' ° *(1997 7 13 12 28 50 508) ° Fdo ÉFX 'do;T;E' 'Σ Expression driver' 'E╜''''' 'æ╜'' ''' 'T╜æ' ° '╕(^/'' ''=T)/0' ° '╕(('':''εT)doif ''B╜evaldd (+/^\'''' ''''=T)╟T'')/2' ° '''E╜ÉEM'' ÉEA T' '╕(0=µ,E)/2' ° '''This is not a valid APL2 expression''' 'æ╜''*''' '╕(''?''⌠╞1╟æ)/2' ° X 'E' '╕2' ° *(1997 7 13 12 28 50 508) ° Fdoif ÉFX 'U╢╜V╢ doif W╢;t╢' 'Σ Rule' '╕(^/~U╢╜V╢)/0' ° X '''U╢╜V╢ doif■ W╢'' ÉEA ''»V╢/W╢''' ° *(1997 9 9 12 50 14 444) ° Fendd ÉFX 'endd' 'Σ end of special feature' '20µ''²'' ╪ ╕(4<µÉLC)/0' ° X 'do' ° *(1997 8 21 13 30 22 444) ° Ferase ÉFX ° 'erase;t;EXIT;GO;BIND;HELP;DISPLAY;ID;DEGREES;RADIANS;MIN;GRADIENT;GRA ° D;DISCLAIMER' 'Σ Erase all global functions and variables' 't╜ÉNL 3' ° 't╜(~t^.εlc,'' '')≡t' 't╜ÉEX(~t[;∞5]^.=''TEACH'')≡t' 't╜ÉNL 2' ° X 't╜ÉEX(~t^.εlc,'' '')≡t' 't╜ÉNL 4' 't╜ÉEX(~t^.εlc,'' '')≡t' ° *(1997 7 27 13 47 41 608) ° Fevaldd ÉFX 'u╜evaldd w;c;n' 'Σ Evaluate direct definition' 'u╜0' ° 'n╜(w∞''Σ'')-1' 'c╜(((n╞w)⌠'':'')Γn╞w),Γ''ΣDD '',(n+1)╟w' ° '╕((1 label╞c)doif ''''''Invalid label'''''')/0' ° '╕((2=µc)doif ''u╜showdd 1╙c'')/0' ° '╕((3=ÉNC╞c)doif ''u╜⌡µÉ╜(╞c),'''' is already defined.'''''')/0' ° '╕((3=µc)doif ''u╜simdd c'')/0' 'c╜(Γ''α∙ aw'')replace■c' ° 'u╜ε''u╜'',((''a''εεc[2 3 4])/''a ''),(╞c),'' w;t;b''' ° 'u╜u(5πc)(''b╜(t╜'',(3πc),'')/'',addquote ''u╜'',4πc)' ° X 'u╜u,''╕(t doif b)/0''(''u╜'',2πc)' 'u╜╧ÉFX u' ° *(1997 7 25 13 27 52 564) ° Fexit ÉFX 'V exit W;T' 'Σ Exit if too many suspended functions' ° '╕(0⌠ÉNC ''V'')/L0 ╪ V╜10' 'L0:╕(V>µÉLC)/0' ° '''There are too many suspended functions''' '''Please enter '',W' ° X '╕' ° *(1997 7 26 12 33 39 536) ° Fget ÉFX 'U╜V get W;t;T;ÉPR' 'Σ Prompt for response from keyboard' ° 'ÉPR╜T╜ÉAV[ÉIO+255] ╪ ╕(0⌠ÉNC ''V'')/L0 ╪ V╜1' 'L0:V╜V╧1' 'æ╜W ╪ t╜æ' ° 'U╜(+/^\t=T)╟t' '╕(''╕''⌠╞U)/L1 ╪ ╕' 'L1:╕V/0' 't╜(U⌠'' '')ΓU' ° X 'U╜(µt),(ΓU),t' ° *(1997 7 28 13 33 8 424) ° Fglobals ÉFX 'globals' 'Σ Initialize useful global variables' ° 'uc╜''ABCDEFGHIJKLMNOPQRSTUVWXYZ''' ° 'lc╜''abcdefghijklmnopqrstuvwxyz''' 'dig╜''1234567890''' ° X 'r╜ÉAV[13+ÉIO]' 'q╜''''''''' ° *(1997 7 3 12 47 6 368) ° Finitialize ÉFX 'initialize;T' 'Σ Initialize workspace' ° '''AT ALL TIMES, TO CONTINUE, PRESS RETURN!'',r' ° '''To see disclaimers enter:'',r,'' disclaimer''' 'do' 'erase' ° 'globals' ° '''Make sure the CAP LOCK light on your keyboard (upper right) is ON!' ° X'' 'endd' ° *(1997 7 27 13 14 33 444) ° Flabel ÉFX 'u╜v label w' ° 'Σ Return 1 if label w does not begin with a cap' ° '╕(0⌠ÉNC ''v'')/L0 ╪ v╜0' 'L0:v╜v╧1 ╪ w╜εw ╪ ╕v/L1 ╪ ╕(u╜0⌠ÉNC w)/0' ° X 'L1:╕(u╜~^/wεlc,uc,dig)/0' 'u╜w[1]εlc,dig' ° XClc 1 26 abcdefghijklmnopqrstuvwxyz ° *(1997 7 13 12 28 55 528) ° Fnon ÉFX 'non;T;RC;ET;R' 'Σ Ignore keyboard entry' 'æ╜'' ''' 'T╜æ' ° '╕(0=µ(T⌠'' '')/T)/0' '(RC ET R)╜ÉEC T' '╕(0=RC)/2' ° X '╕((1=RC)doif ''R'')/2' '╕2' ° *(1997 7 13 12 28 55 528) ° Fnotb ÉFX 'u╜notb w' 'Σ Remove trailing blanks' ° '╕((1<╧w)doif ''u╜notb■ w'')/0' '╕((1<µµw)doif ''u╜πnotb Γ[2]w'')/0' ° X 'u╜(1-(,'' ''⌠Φw)∞1)╟w' ° *(1996 4 6 16 11 57 440) ° Fpause ÉFX 'V pause W;T' ° 'Σ Pause, then print W V spaces right and return' ° X 'T╜(0=ÉNC ''V'')doif ''V╜6''' 'do' '(Vµ'' ''),W' 'do' ° *(1997 7 27 12 55 6 496) ° Fproblems ÉFX 'problems' 'Σ Problems' ° '''That is all for this lesson. Remember, if you want to practice,''' ° '''and plan to use direct definitions, be sure to first enter GO.''' ° '''Direct definitions will then be accepted. To exit GO, enter EXIT.'' ° ,r' ° '''To erase a previously defined DIRECT DEFINITION FUNCTION, enter'',r ° ' ''' )ERASE functionname'',r' ° X '''WARNING! do not use )ERASE on other labels.'',r' ° XCq 0 ' ° XCr 0 ° *(1997 7 13 12 28 56 532) ° Freplace ÉFX 'u╜v replace u;i;r;s' 'Σ Replace elements in v in u' ° 'i╜Γ∞µu' 's╜2πv╜(v⌠'' '')Γv' 'i╜⌡r╜i⌡■Γ[1]u°.=╞v' ° X 'u[(εi)/εr]╜s[(εi)/εi⌡■∞µs]' ° *(1997 7 13 12 28 56 532) ° Fround ÉFX 'U╜V round W' 'Σ Half adjust to V th decimal' ° X 'U╜(╛0.5+W⌡10*V)÷10*V' ° *(1997 7 13 12 28 57 536) ° Fshow ÉFX '╢V show ╢W;╢T;╢B' 'Σ Display and execute ╢W' ° '╢T╜(0=ÉNC ''╢V'')doif ''╢V╜0''' ° '╕((0=╧╢W)doif ''show ╢W,'''' '''''')/0' ° '╕((1<╧╢W)doif ''╢V show ■╢W'')/0' ''' '',╢W' ° X '╕((╢V^'':''ε╢W)doif ''╢T╜evaldd ╢W'')/L0' '''ÉEM'' ÉEA ╢W' 'L0:do' ° *(1997 7 13 12 28 57 536) ° Fshowdd ÉFX 'u╜showdd w;a;b;c;r' ° 'Σ Display a direct definition function' ° '╕((1=╧w)doif ''u╜showdd Γw'')/u╜0' ° '╕((3⌠ÉNC╞w)doif ''(ε╞w),'''' is not a function'''''')/0' ° 'c╜Γ[2]ÉCR╞w' 'c╜notb(2╞c),(Γ''aw α∙'')replace■2╟c' ° '╕((~''ΣDD''╧3╞2πc)doif ''''''Not a direct definition function'''''')/ ° 0' 'u╜1' 'b╜('' ''⌠╞c)Γ╞c' 'a╜'' ''' 'r╜2╟3πc' ° '╕((3=µc)doif ''a,(╞w),'''':'''',r,(3<µ2πc)/'''' Σ'''',3╟2πc'')/0' ° 'a╜a,(╞w),'':'',(2╟5πc),'':''' 'b╜(+\r=''('')-+\r='')''' 'b╜b∞0' ° X 'a╜a,(²3╟(b-1)╞3╟r),'':'',2╟»(b+2)╟r' 'a,(3<µ2πc)/'' Σ'',3╟2πc' ° *(1997 7 13 12 28 57 536) ° Fshowfn ÉFX 'U╜V showfn W;F;N;T;ÉIO' 'Σ Simulate ╖W[É]' ° 'T╜(0=ÉNC ''V'')doif ''V╜0''' 'ÉIO╜0' ° 'U╜r,'' '',''╖'',W,''[É]'',(╞V)╞''╖''' 'N╜1╞µF╜ÉCR W' 'N╜«∞N' ° 'N╜(N⌠'' '')ΓN' 'F╜(π''['',■N,■Γ''] ''),F' ° 'T╜(1<µ,V)doif ''F╜F[1╟V;]'' ''U╜''''''''''' 'U╜²1╟U,r,,F,r' ° X 'U╜((-+/^\'' ''=ΦU)╟U),('' ╖'')[╞V],r' ° *(1997 7 13 12 28 58 540) ° Fsimdd ÉFX 'u╜simdd w;e' 'Σ Direct definition mode' 'u╜0' ° '╕((0⌠ÉNC╞w)doif ''''''Already defined'''''')/0' 'e╜''α''ε2πw' ° 'w[2]╜Γ''u╜'',''α∙ aw'' replace 2πw' 'w╜w[1 3 2]' ° X 'w[1]╜Γε''u╜'',(e/''a ''),w[1],'' w''' 'u╜╧ÉFX w' ° *(1992 6 3 9 59 17 424) ° Ftab ÉFX 'U╜V tab W;T;A;B;C;D;E;F;G;M;ÉPW' 'Σ Tabulate list W' ° 'T╜(0=ÉNC ''V'')doif ''V╜0''' 'M╜''Invalid data for tabulation''' ° 'V╜4╞V' 'ÉPW╜130╛30⌐G╜V[2]+79⌡V[2]=0' ° 'L1:╕((1<╧W)doif ''''''W╜∞0'''' ÉEA ''''W╜πW'''''')/L1' ° '╕(((0=µεW)δ2<µµW)doif ''U╜(~V╧4╞0)/M'')/0' ° 'T╜(1≥µµU╜«W)doif ''U╜πW╜(U⌠'''' '''')ΓU''' ° 'T╜(0<V[1])doif ''U╜(«(Φ1,╞µW)µ(V[3]µ'''' ''''),∞(╞µW)-V[3]),'''' '''' ° ,U''' '╕(G<30)/0' 'T╜(F╜µεV[4])+C╜1╟B╜µA╜(V[3],0)╟U' ° 'T╜⌐(1╞B)÷1⌐╛(ÉPW+F)÷T' 'U╜(E╜(V[3],C)╞U),[1](B╜T,1╟B)╞A' ° '''D╜εV[4]'' ÉEA ''D╜ÉAV[εV[4]+33⌡V[4]=0]''' 'L0:A╜(T,0)╟A' ° X '╕(0=1╞µA)/0' 'U╜U,(((T+V[3]),µD)µD),E,[1]B╞A' '╕L0' ° *(1997 7 13 12 28 59 544) ° Ftest ÉFX 'U╜V test W;P' ° 'Σ Describe problem in W, (correct answer in V)' 'U╜2' 'L1:W' ° 'É╜'' ''' 'P╜æ' '''╕L0'' ÉEA ''P╜»P''' '╕(V╧P)/0' ° X 'L0:╕(0=U╜U-1)/0' '''Incorrect. Try again''' '╕L1' ° XCuc 1 26 ABCDEFGHIJKLMNOPQRSTUVWXYZ ° *(1997 8 21 13 15 13 348) ° Fupdate ÉFX 'update' ° '''This version of APL2 lessons has been updated in the summer of 1997 ° ''' ° '''Any subsequent updates will be announced to the APL news group.''' ° '''You may also e-mail me at jizba@kaiwan.com for additional details.' ° '' '''Any constructive comments will be welcome.'',r' ° '''For an update of these lessons, mail a self addressed stamped''' ° '''floppy disk mailer with two HD 3.5 inch blank floppies to:'',r' ° ''' Zdenek V. Jizba''' ° ''' 1341 Rebecca Dr.''' ° X ''' La Habra, CA 90631''' ° *(1996 4 6 16 11 57 440) ° Fvec ÉFX 'U╜V vec W;T' 'Σ Draw 2d vector in plane' ° 'T╜(0=ÉNC ''V'')doif ''V╜''''''''''' ° 'U╜r,'' '',W[3],r,''2- +'',W[2],r '': °'',r,''1- °'',r' ° X 'U╜U,'' : °'',r,''0|....|....|.'',V,,'' '',W[1],r,'' 0 1 2'',r' °