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 XXX TO THE FACTORIAL OF 10; XXX := 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,N; 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 FAC 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); SCALAR F1,F2,G1,G2; 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; 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; COMMENT LEAVING SUCH REPLACEMENTS ACTIVE WOULD SLOW DOWN SUBSEQUENT COMPUTATION; 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*XXX; ZZ:= SOLVE (XX,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; 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 ",!*ANS; 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; 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 TRACE ALGEBRA IS DONE; WRITE "CXN = ",!*ANS; COMMENT END OF SECOND PHYSICS EXAMPLE; COMMENT THE FOLLOWING RATHER LONG PROGRAM IS A COMPLETE ROUTINE FOR CALCULATING THE RICCI SCALAR. IT WAS DEVELOPED IN COLLABORATION WITH DAVID BARTON AND JOHN FITCH; COMMENT FIRST WE INHIBIT DIAGNOSTIC MESSAGE PRINTING AND THE PRINTING OF ZERO ELEMENTS OF ARRAYS; OFF MSG$ ON NERO$ COMMENT HERE WE INTRODUCE THE COVARIANT AND CONTRAVARIANT METRICS; ARRAY GG(3,3),H(3,3),X(3)$ FOR I:=0:3 DO FOR J:=0:3 DO GG(I,J):=H(I,J):=0$ 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)$ IF I UNEQ J LET DF(P1(X(I)),X(J))=0, DF(Q1(X(I)),X(J))=0; COMMENT GENERATE CHRISTOFFEL SYMBOLS AND STORE IN ARRAYS CS1 AND CS2; ARRAY CS1(3,3,3)$ FOR I:=0:3 DO FOR J:=I:3 DO 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$ ARRAY CS2(3,3,3)$ FOR I:= 0:3 DO FOR J:=I:3 DO 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)$ COMMENT NOW CALCULATE THE DERIVATIVES OF THE CHRISTOFFEL SYMBOLS AND STORE IN DC2(I,J,K,L); ARRAY DC2(3,3,3,3)$ FOR I:=0:3 DO FOR J:=I:3 DO FOR K:=0:3 DO FOR L:=0:3 DO DC2(J,I,K,L) := DC2(I,J,K,L):=DF(CS2(I,J,K),X(L))$ COMMENT NOW STORE THE SUMS OF PRODUCTS OF THE CS2 IN SPCS2; ARRAY SPCS2(3,3,3,3)$ FOR I:=0:3 DO FOR J:=I:3 DO FOR K:=0:3 DO FOR L:=0:3 DO SPCS2(J,I,K,L) := SPCS2(I,J,K,L) := FOR P := 0:3 SUM CS2(P,L,K)*CS2(I,J,P)$ 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)*(DC2(K,J,Q,L)-DC2(J,L,Q,K) +SPCS2(K,J,Q,L)-SPCS2(L,J,Q,K))$ R(I,J,L,K) := 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)$ R(L,K,I,J) := 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 FINALLY COMPUTE AND PRINT THE RICCI SCALAR; R := FOR I:= 0:3 SUM FOR J:= 0:3 SUM H(I,J)*RICCI(I,J); END OF RICCI TENSOR AND SCALAR CALCULATION; COMMENT END OF ALL EXAMPLES; END;