reduce-historical/r30/reduce.tst

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COMMENT SOME EXAMPLES OF THE  F O R  STATEMENT;

COMMENT SUMMING THE SQUARES OF THE EVEN POSITIVE INTEGERS
	THROUGH 50;

FOR I:=2 STEP 2 UNTIL 50 SUM I**2;

COMMENT TO SET  W  TO THE FACTORIAL OF 10;

W := FOR I:=1:10 PRODUCT I;

COMMENT ALTERNATIVELY, WE COULD SET THE ELEMENTS A(I) OF THE
	ARRAY  A  TO THE FACTORIAL OF I BY THE STATEMENTS;

ARRAY A(10);
A(0):=1$
FOR I:=1:10 DO A(I):=I*A(I-1);

COMMENT THE ABOVE VERSION OF THE F O R STATEMENT DOES NOT RETURN
	AN ALGEBRAIC VALUE, BUT WE CAN NOW USE THESE ARRAY
	ELEMENTS AS FACTORIALS IN EXPRESSIONS, E. G.;

1+A(5);

COMMENT WE COULD HAVE PRINTED THE VALUES OF EACH A(I)
	AS THEY WERE COMPUTED BY REPLACING THE F O R STATEMENT BY;

FOR I:=1:10 DO WRITE A(I):= I*A(I-1);

COMMENT ANOTHER WAY TO USE FACTORIALS WOULD BE TO INTRODUCE AN
OPERATOR  FAC  BY AN INTEGER PROCEDURE AS FOLLOWS;

INTEGER PROCEDURE FAC (N);
   BEGIN INTEGER M;
	M:=1;
    L1:	IF N=0 THEN RETURN M;
	M:=M*N;
	N:=N-1;
	GO TO L1
   END;

COMMENT WE CAN NOW USE  FAC  AS AN OPERATOR IN EXPRESSIONS,
E. G.;

Z**2+FAC(4)-2*FAC 2*Y;

COMMENT NOTE IN THE ABOVE EXAMPLE THAT THE PARENTHESES AROUND
THE ARGUMENTS OF  FAC  MAY BE OMITTED SINCE IT IS A UNARY OPERATOR;

COMMENT THE FOLLOWING EXAMPLES ILLUSTRATE THE SOLUTION OF SOME
	COMPLETE PROBLEMS;

COMMENT THE F AND G SERIES (REF  SCONZO, P., LESCHACK, A. R. AND    
         TOBEY, R. G., ASTRONOMICAL JOURNAL, VOL 70 (MAY 1965);

DEPS:= -SIG*(MU+2*EPS)$
DMU:= -3*MU*SIG$
DSIG:= EPS-2*SIG**2$
F1:= 1$
G1:= 0$
 
FOR I:= 1:8 DO 
 BEGIN
   F2:= -MU*G1 + DEPS*DF(F1,EPS) + DMU*DF(F1,MU) + DSIG*DF(F1,SIG)$
   WRITE "F(",I,") := ",F2;
   G2:= F1 + DEPS*DF(G1,EPS) + DMU*DF(G1,MU) + DSIG*DF(G1,SIG)$
   WRITE "G(",I,") := ",G2;
   F1:=F2$
   G1:=G2
  END;

COMMENT A PROBLEM IN FOURIER ANALYSIS;

FOR ALL X,Y LET COS(X)*COS(Y)= (COS(X+Y)+COS(X-Y))/2,  
		COS(X)*SIN(Y)= (SIN(X+Y)-SIN(X-Y))/2,  
		SIN(X)*SIN(Y)= (COS(X-Y)-COS(X+Y))/2,  
		COS(X)**2= (1+COS(2*X))/2,
		SIN(X)**2= (1-COS(2*X))/2;


FACTOR COS,SIN;


ON LIST;


(A1*COS(WT)+ A3*COS(3*WT)+ B1*SIN(WT)+ B3*SIN(3*WT))**3;

COMMENT END OF FOURIER ANALYSIS EXAMPLE; 

OFF LIST;
FOR ALL X,Y CLEAR COS X*COS Y, COS X*SIN Y, SIN X*SIN Y,
	          COS(X)**2,SIN(X)**2;


COMMENT LEAVING SUCH REPLACEMENTS ACTIVE WOULD SLOW DOWN SUBSEQUENT
	COMPUTATION;

COMMENT THE FOLLOWING PROGRAM, WRITTEN IN  COLLABORATION  WITH  DAVID
BARTON  AND  JOHN  FITCH,  SOLVES A PROBLEM IN GENERAL RELATIVITY. IT
WILL COMPUTE THE EINSTEIN TENSOR FROM ANY GIVEN METRIC;

ON NERO;

COMMENT HERE WE INTRODUCE THE COVARIANT AND CONTRAVARIANT METRICS;

OPERATOR P1,Q1,X;
ARRAY GG(3,3),H(3,3)$
GG(0,0):=E**(Q1(X(1)))$
GG(1,1):=-E**(P1(X(1)))$
GG(2,2):=-X(1)**2$
GG(3,3):=-X(1)**2*SIN(X(2))**2$
FOR I:=0:3 DO  H(I,I):=1/GG(I,I)$

COMMENT GENERATE CHRISTOFFEL SYMBOLS AND STORE IN ARRAYS CS1 AND CS2;

ARRAY CS1(3,3,3),CS2(3,3,3)$
FOR I:=0:3 DO FOR J:=I:3 DO BEGIN
    FOR K:=0:3 DO 
       CS1(J,I,K) := CS1(I,J,K):=(DF(GG(I,K),X(J))+DF(GG(J,K),X(I))
       -DF(GG(I,J),X(K)))/2;
        FOR K:=0:3 DO CS2(J,I,K):= CS2(I,J,K) := FOR P := 0:3 
				SUM H(K,P)*CS1(I,J,P) END;

COMMENT NOW COMPUTE THE RIEMANN TENSOR AND STORE IN R(I,J,K,L);

ARRAY R(3,3,3,3)$
FOR I:=0:3 DO FOR J:=I+1:3 DO FOR K:=I:3 DO
   FOR L:=K+1:IF K=I THEN J ELSE 3 DO BEGIN
	R(J,I,L,K) := R(I,J,K,L) := FOR Q := 0:3 
		SUM GG(I,Q)*(DF(CS2(K,J,Q),X(L))-DF(CS2(J,L,Q),X(K))
		+ FOR P:=0:3 SUM (CS2(P,L,Q)*CS2(K,J,P)
			-CS2(P,K,Q)*CS2(L,J,P)))$
	LET R(I,J,L,K) = -R(I,J,K,L), R(J,I,K,L)= -R(I,J,K,L);
	IF I=K AND J<=L THEN GO TO A$
	R(K,L,I,J) := R(L,K,J,I) := R(I,J,K,L)$
	LET R(L,K,I,J) = -R(I,J,K,L), R(K,L,J,I)= -R(I,J,K,L);
 A: END$

COMMENT NOW COMPUTE AND PRINT THE RICCI TENSOR;

ARRAY RICCI(3,3)$
FOR I:=0:3 DO FOR J:=0:3 DO  
    WRITE RICCI(J,I) := RICCI(I,J) := FOR P := 0:3 SUM FOR Q := 0:3
					SUM H(P,Q)*R(Q,I,P,J);

COMMENT NOW COMPUTE AND PRINT THE RICCI SCALAR;

RS := FOR I:= 0:3 SUM FOR J:= 0:3 SUM H(I,J)*RICCI(I,J);

COMMENT FINALLY COMPUTE AND PRINT THE EINSTEIN TENSOR;

ARRAY EINSTEIN(3,3);

FOR I:=0:3 DO FOR J:=0:3 DO
	 WRITE EINSTEIN(I,J):=RICCI(I,J)-RS*GG(I,J)/2;

COMMENT END OF EINSTEIN TENSOR PROGRAM;

CLEAR GG,H,CS1,CS2,R,RICCI,EINSTEIN;

COMMENT AN EXAMPLE USING THE MATRIX FACILITY;

MATRIX XX,YY;

LET XX= MAT((A11,A12),(A21,A22)),
   YY= MAT((Y1),(Y2));

2*DET XX - 3*W;

ZZ:= XX**(-1)*YY;

1/XX**2;

COMMENT END OF MATRIX EXAMPLES;

COMMENT THE FOLLOWING EXAMPLES WILL FAIL UNLESS THE FUNCTIONS 
        NEEDED FOR PROBLEMS IN HIGH ENERGY PHYSICS HAVE BEEN LOADED;

COMMENT A PHYSICS EXAMPLE;
ON DIV; COMMENT THIS GIVES US OUTPUT IN SAME FORM AS BJORKEN AND DRELL;

MASS KI= 0, KF= 0, PI= M, PF= M;

VECTOR EI,EF;

MSHELL KI,KF,PI,PF; 
LET PI.EI= 0, PI.EF= 0, PI.PF= M**2+KI.KF, PI.KI= M*K,PI.KF= 
    M*KP, PF.EI= -KF.EI, PF.EF= KI.EF, PF.KI= M*KP, PF.KF=    
    M*K, KI.EI= 0, KI.KF= M*(K-KP), KF.EF= 0, EI.EI= -1, EF.EF=
    -1; 
OPERATOR GP;
FOR ALL P LET GP(P)= G(L,P)+M;
COMMENT THIS IS JUST TO SAVE US A LOT OF WRITING;
GP(PF)*(G(L,EF,EI,KI)/(2*KI.PI) + G(L,EI,EF,KF)/(2*KF.PI)) 
  * GP(PI)*(G(L,KI,EI,EF)/(2*KI.PI) + G(L,KF,EF,EI)/(2*KF.PI)) $    
WRITE "THE COMPTON CROSS-SECTION IS ",WS;
COMMENT END OF FIRST PHYSICS EXAMPLE; 

OFF DIV;

COMMENT ANOTHER PHYSICS EXAMPLE;
FACTOR MM,P1.P3;
INDEX X1,Y1,Z;
MASS P1=MM,P2=MM,P3= MM,P4= MM,K1=0;
MSHELL P1,P2,P3,P4,K1;
VECTOR Q1,Q2; 
OPERATOR GA,GB;
FOR ALL P LET GA(P)=G(LA,P)+MM, GB(P)= G(LB,P)+MM; 
GA(-P2)*G(LA,X1)*GA(-P4)*G(LA,Y1)* (GB(P3)*G(LB,X1)*GB(Q1)  
    *G(LB,Z)*GB(P1)*G(LB,Y1)*GB(Q2)*G(LB,Z)   +   GB(P3)     
    *G(LB,Z)*GB(Q2)*G(LB,X1)*GB(P1)*G(LB,Z)*GB(Q1)*G(LB,Y1))$ 
LET Q1=P1-K1, Q2=P3+K1; 
COMMENT IT IS USUALLY FASTER TO MAKE SUCH SUBSTITUTIONS AFTER ALL THE
	TRACE ALGEBRA IS DONE;
WRITE "CXN =",WS;

COMMENT END OF SECOND PHYSICS EXAMPLE; 

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END;