Info 3.6.

Chapter 3.6

Programming

of

Numerical Algorithms

2Info 3.6.

Numerical Algorithms

• Generic procedures to – tabulate, plot, ... a given function– to find the zeros of a given function– to compute the derivative of a given function– to integrate a given function

• How is the given function defined ?– By a function procedure– Passed as a parameter to the generic procedures

3Info 3.6.

TYPES

• Simple Types– Ordinal Types– REAL Type– POINTER Type

• Structured Types– ARRAY– RECORD– SET– PROCEDURE

4Info 3.6.

PROCEDURE TypeTYPE RFunction = PROCEDURE(REAL):REAL;

PROCEDURE SIN(x : REAL):REAL;BEGIN RETURN VAL(REAL,Sin(VAL(LONGREAL,x)))END SIN;PROCEDURE F1(x:REAL) : REAL;BEGIN RETURN x - VAL(REAL,Exp(VAL(LONGREAL,1/x)))END F1;PROCEDURE F2(x:REAL) : REAL;BEGIN RETURN x*x*x/3.0 - 3.5 * x*x + 10.0 * x - 5.0;END F2;

5Info 3.6.

Tabulating a function

PROCEDURE Tab(F : RFunction; First, Last : REAL; NPoints : CARDINAL); VAR x,Step : REAL;BEGIN IF Last > First THEN x := First; Step := (Last - First) / FLOAT(NPoints-1); REPEAT WrReal(x,15,15); WrStr(" : "); WrReal(F(x),15,15); WrLn; x := x + Step UNTIL x > Last; END (* IF *)END Tab;

6Info 3.6.

Plot Procedure

Procedure HeaderPROCEDURE PlotRF(F : RFunction; First, Last, Min, Max : REAL; NXPoints, NYPoints : CARDINAL);

(* F : Y = F(X) is the function to be *)(* plotted by PlotRF *)(* First : First value of X for the plot *)(* Last : Last value of X for the plot *)(* Min,Max : Extreme Y values of the plot *)(* NXPoints : Number of X points on screen *)(* NYPoints : Number of Y points on screen *)

7Info 3.6.

Plot Procedure

Core CodePROCEDURE PlotRF(F : RFunction; First, Last, Min, Max : REAL; NXPoints, NYPoints : CARDINAL);BEGIN XStep := (Last-First)/FLOAT(NXPoints); YStep := (Max-Min)/FLOAT(NYPoints); IF Init(0,0,NXPoints,NYPoints) THEN FOR x := 0 TO NXPoints DO YValue := (F(XStep*FLOAT(x)+First)-Min)/YStep; y := NYPoints - TRUNC(YValue); Plot(x,y,_clrLIGHTRED) END (* FOR *) END; (* IF Init *)END PlotRF;

8Info 3.6.

Plot Procedure

Adding the X axis

FOR x := 0 TO NXPoints DO z := TRUNC((0.0 - Min)/YStep); Plot(x,z,_clrGREEN); ...END (* FOR *)

9Info 3.6.

Plot ProcedureTesting for reasonable

ranges BEGIN IF Last > First THEN ... FOR x := 0 TO NXPoints DO YValue := (F(XStep*FLOAT(x)+First)-Min)/YStep; IF (YValue >= 0) AND (YValue <= FLOAT(NYPoints) THEN y := NYPoints - TRUNC(YValue); Plot(x,y,_clrLIGHTRED) END (* IF in plot range test *) END (* FOR *) ... END (* IF Last > First *) END PlotRF;

10Info 3.6.

Plot ProcedurePROCEDURE PlotRF(F : RFunction; First, Last, Min, Max : REAL;

NXPoints, NYPoints : CARDINAL); VAR x,y,z : CARDINAL; YValue : REAL; XStep, YStep : REAL;BEGIN IF Last > First THEN XStep := (Last-First)/FLOAT(NXPoints); YStep := (Max-Min)/FLOAT(NYPoints); IF Init(0,0,NXPoints,NYPoints) THEN FOR x := 0 TO NXPoints DO z := TRUNC((0.0 - Min)/YStep); Plot(x,z,_clrGREEN); YValue := (F(XStep*FLOAT(x)+First)-Min)/YStep; IF (YValue >= 0) AND (YValue <= FLOAT(NYPoints)) THEN y := NYPoints - TRUNC(YValue); Plot(x,y,_clrLIGHTRED) END (* IF in plot range test *) END (* FOR *) END; (* IF Init *) END (* IF Last > First *)END PlotRF;

11Info 3.6.

Newton's Algorithmfor finding a root of a

function

x0x1

|X1-X| < |X0-X|

x

12Info 3.6.

Procedure NEWTONParameters

PROCEDURE Newton (f :RFunction (*Function *); df:RFunction (*First derivative *);GuessedRoot:REAL (*Initial guess *);VAR Root:REAL (*Computed root *); tol:REAL (*Desired precision *); MaxIter:CARDINAL (*Max.Nbr.iterations*);VAR Iter:CARDINAL (*Act.Nbr.iterations*);VAR State:IterState (*Procedure state *));

13Info 3.6.

Procedure NEWTONState VariableIterState = (iterating,rootfound,

toomanyiter,toonearzero,tooflat); Iter := 0; State := iterating;REPEAT IF Iter = MaxIter THEN State := toomanyiter ELSIF ... THEN State := toonearzero ELSIF ... THEN State := tooflat ELSE Iter := Iter + 1; ... IF ... < tol THEN State := rootfound END ENDUNTIL State # iterating;

14Info 3.6.

Procedure NEWTON

toonearzero - tooflatCONST NearZero = 1.0E-20; ...BEGIN ... REPEAT ... ELSIF ABS(Root) < NearZero THEN State := toonearzero ELSIF ABS(dfx) < NearZero THEN State := tooflat ... UNTIL State # iterating;END Newton;

15Info 3.6.

Procedure NEWTON

rootfound Iter := 0; Root := GuessedRoot; fx := f(Root); REPEAT dfx := df(Root); ... Iter := Iter + 1; OldRoot := Root; Root := Root - fx/dfx; fx := f(Root); IF ABS((Root-OldRoot)/OldRoot) < tol THEN State := rootfound END UNTIL State # iterating;

16Info 3.6.

Procedure NEWTONPROCEDURE Newton (f,df:RFunction;GuessedRoot:REAL; VAR Root:REAL;tol:REAL;

MaxIter:CARDINAL;VAR Iter:CARDINAL;VAR State:IterState); CONST NearZero = 1.0E-20; VAR fx,dfx,OldRoot : REAL;BEGIN Root := GuessedRoot; fx := f(Root); Iter := 0; State := iterating; REPEAT dfx := df(Root); IF Iter = MaxIter THEN State := toomanyiter ELSIF ABS(Root) < NearZero THEN State := toonearzero ELSIF ABS(dfx) < NearZero THEN State := tooflat ELSE Iter := Iter + 1; OldRoot := Root; Root := Root - fx/dfx; fx := f(Root); IF ABS((Root-OldRoot)/OldRoot) < tol THEN State := rootfound END END UNTIL State # iterating;END Newton;