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$RUN NEW:LISP SCARDS=*SOURCE*+*MSOURCE* 
VERBOS (NIL) PRBUFFER (NIL)
OPEN (WATF:RDC2CHKPT SYSFILE INPUT)
RESTORE (WATF:RDC2CHKPT)
BEGIN NIL

     REDUCE2(15-SEP-72 (UM 1-JUNE-73)) ... 
 
 
 
     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; 
 
     22100 
 
 
     COMMENT TO SET XXX TO THE FACTORIAL OF 10; 
 
 
 
     XXX:=FOR I:=1:10 PRODUCT I; 
 
     XXX:=3628800 
 
 
     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); 
 
     121 
 
 
     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); 
 
     A(1):=1 
 
     A(2):=2 
 
     A(3):=6 
 
     A(4):=24 
 
     A(5):=120 
 
     A(6):=720 
 
     A(7):=5040 
 
     A(8):=40320 
 
     A(9):=362880 
 
     A(10):=3628800 
 
 
     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; 
 
                2 
      - (4*Y - Z  - 24) 
 
 
     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; 
 
 
     F(1) := 0 
 
     G(1) := 1 
 
     F(2) :=  - MU 
 
     G(2) := 0 
 
     F(3) := 3*MU*SIG 
 
     G(3) :=  - MU 
 
                            2 
     F(4) := MU*(MU - 15*SIG  + 3*EPS) 
 
     G(4) := 6*MU*SIG 
 
                                     2 
     F(5) :=  - 15*MU*SIG*(MU - 7*SIG  + 3*EPS) 
 
                            2 
     G(5) := MU*(MU - 45*SIG  + 9*EPS) 
 
                      2             2                      4          2             2 
     F(6) :=  - MU*(MU  - 210*MU*SIG  + 24*MU*EPS + 945*SIG  - 630*SIG *EPS + 45*EPS ) 
 
                                      2 
     G(6) :=  - 30*MU*SIG*(MU - 14*SIG  + 6*EPS) 
 
                          2            2                      4          2             2 
     F(7) := 63*MU*SIG*(MU  - 50*MU*SIG  + 14*MU*EPS + 165*SIG  - 150*SIG *EPS + 25*EPS ) 
 
                      2             2                       4           2              2 
     G(7) :=  - MU*(MU  - 630*MU*SIG  + 54*MU*EPS + 4725*SIG  - 3150*SIG *EPS + 225*EPS ) 
 
                   3          2    2         2                   4               2                  2             6 
     F(8) := MU*(MU  - 2205*MU *SIG  + 117*MU *EPS + 51975*MU*SIG  - 24570*MU*SIG *EPS + 1107*MU*EPS  - 135135*SIG  + 
 
                           4                2    2           3 
                 155925*SIG *EPS - 42525*SIG *EPS  + 1575*EPS ) 
 
                           2             2                      4          2             2 
     G(8) := 126*MU*SIG*(MU  - 100*MU*SIG  + 24*MU*EPS + 495*SIG  - 450*SIG *EPS + 75*EPS ) 
 
 
     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; 
 
               3   3 
     (4*SIN(WT) *B1 
 
                        2 
       + 3*SIN(WT)*(2*B3 *B1 
 
              2 
       - B3*B1 
 
              2 
       + B3*A1 
 
                2 
       + 2*B1*A3 
 
       - 2*B1*A3*A1 
 
              2 
       + B1*A1 ) 
 
                          2 
       + 3*SIN(9*WT)*B3*A3 
 
                        2 
       - 3*SIN(7*WT)*(B3 *B1 
 
       - 2*B3*A3*A1 
 
              2 
       - B1*A3 ) 
 
                        2 
       + 3*SIN(5*WT)*(B3 *B1 
 
              2 
       - B3*B1 
 
       + 2*B3*A3*A1 
 
              2 
       + B3*A1 
 
              2 
       - B1*A3 
 
       + 2*B1*A3*A1) 
 
                    3   3 
       + 4*SIN(3*WT) *B3 
 
                             2 
       + 3*SIN(3*WT)*(2*B3*B1 
 
              2 
       + B3*A3 
 
                2 
       + 2*B3*A1 
 
              2 
       + B1*A1 ) 
 
                  3   3 
       + 4*COS(WT) *A1 
 
                        2 
       + 3*COS(WT)*(2*B3 *A1 
 
       + 2*B3*B1*A1 
 
           2 
       - B1 *A3 
 
           2 
       + B1 *A1 
 
             2 
       + 2*A3 *A1 
 
              2 
       + A3*A1 ) 
 
                       2 
       - 3*COS(9*WT)*B3 *A3 
 
                        2 
       - 3*COS(7*WT)*(B3 *A1 
 
       + 2*B3*B1*A3 
 
           2 
       - A3 *A1) 
 
                        2 
       - 3*COS(5*WT)*(B3 *A1 
 
       - 2*B3*B1*A3 
 
       + 2*B3*B1*A1 
 
           2 
       + B1 *A3 
 
           2 
       - A3 *A1 
 
              2 
       - A3*A1 ) 
 
                    3   3 
       + 4*COS(3*WT) *A3 
 
                        2 
       + 3*COS(3*WT)*(B3 *A3 
 
             2 
       + 2*B1 *A3 
 
           2 
       - B1 *A1 
 
                2 
       + 2*A3*A1 )) 
 
     /4 
 
 
     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; 
 
     2*(A22*A11 - A21*A12 - 5443200) 
 
 
     ZZ:=SOLVE(XX,YY); 
 
     ZZ(1,1)=(Y1*A22 - Y2*A12)/(A22*A11 - A21*A12) 
 
 
     ZZ(2,1)=( - (Y1*A21 - Y2*A11))/(A22*A11 - A21*A12) 
 
 
 
 
     1/XX**2; 
 
                  2                2    2                          2    2 
     MAT(1,1)=(A22  + A21*A12)/(A22 *A11  - 2*A22*A21*A12*A11 + A21 *A12 ) 
 
 
                                       2    2                          2    2 
     MAT(1,2)=( - A12*(A22 + A11))/(A22 *A11  - 2*A22*A21*A12*A11 + A21 *A12 ) 
 
 
                                       2    2                          2    2 
     MAT(2,1)=( - A21*(A22 + A11))/(A22 *A11  - 2*A22*A21*A12*A11 + A21 *A12 ) 
 
 
                            2      2    2                          2    2 
     MAT(2,2)=(A21*A12 + A11 )/(A22 *A11  - 2*A22*A21*A12*A11 + A21 *A12 ) 
 
 
 
 
     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; 
 
                                       (-1)   (-1)   2               2              2 
     THE COMPTON CROSS-SECTION IS 1/2*K    *KP    *(K  + 4*K*KP*EF.EI  - 2*K*KP + KP ) 
 
 
     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; 
 
                4             4                        2      2        2                                     2 
     CXN = 32*MM *P3.P1 - 8*MM *(P3.K1 - P1.K1) - 16*MM *P3.P1  - 16*MM *P3.P1*(P4.P2 - P3.K1 + P1.K1) + 8*MM *(P4.P2* 
 
           P3.K1 - P4.P2*P1.K1 - 2*P4.K1*P2.K1) + 8*P3.P1*(2*P4.P3*P2.P1 - P4.P3*P2.K1 + 2*P4.P1*P2.P3 + P4.P1*P2.K1 
 
            - P4.K1*P2.P3 + P4.K1*P2.P1) + 8*(2*P4.P3*P2.P3*P1.K1 - P4.P3*P2.P1*P3.K1 + P4.P3*P2.P1*P1.K1 - P4.P1*P2. 
 
           P3*P3.K1 + P4.P1*P2.P3*P1.K1 - 2*P4.P1*P2.P1*P3.K1) 
 
 
     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~=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&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); 
 
                                                 ( - Q1(X(1)))         ( - (P1(X(1)) - Q1(X(1)))) 
     RICCI(0,0):=RICCI(0,0):=(4*R(0,0,0,0)*X(1)*E              + X(1)*E                          *DF(P1(X(1)),X(1))*DF 
 
                             ( - (P1(X(1)) - Q1(X(1))))                  2           ( - (P1(X(1)) - Q1(X(1)))) 
     (Q1(X(1)),X(1)) - X(1)*E                          *DF(Q1(X(1)),X(1))  - 2*X(1)*E                          *DF(Q1( 
 
                           ( - (P1(X(1)) - Q1(X(1)))) 
     X(1)),X(1),X(1)) - 4*E                          *DF(Q1(X(1)),X(1)))/(4*X(1)) 
 
                                             ( - P1(X(1)))               ( - Q1(X(1))) 
     RICCI(1,0):=RICCI(0,1):= - (R(1,0,1,1)*E              - R(0,0,0,1)*E             ) 
 
                                                              2  ( - Q1(X(1)))       2 
     RICCI(2,0):=RICCI(0,2):=( - (R(2,0,2,2) - R(0,0,0,2)*X(1) *E             ))/X(1) 
 
                                       2                2  ( - Q1(X(1)))                         2     2 
     RICCI(3,0):=RICCI(0,3):=(SIN(X(2)) *R(0,0,0,3)*X(1) *E              - R(3,0,3,3))/(SIN(X(2)) *X(1) ) 
 
                                             ( - P1(X(1)))               ( - Q1(X(1))) 
     RICCI(0,1):=RICCI(1,0):= - (R(1,1,1,0)*E              - R(0,1,0,0)*E             ) 
 
                                                     ( - P1(X(1))) 
     RICCI(1,1):=RICCI(1,1):=( - (4*R(1,1,1,1)*X(1)*E              + X(1)*DF(P1(X(1)),X(1))*DF(Q1(X(1)),X(1)) - X(1)* 
 
                      2 
     DF(Q1(X(1)),X(1))  - 2*X(1)*DF(Q1(X(1)),X(1),X(1)) + 4*DF(P1(X(1)),X(1))))/(4*X(1)) 
 
                                                              2  ( - P1(X(1)))       2 
     RICCI(2,1):=RICCI(1,2):=( - (R(2,1,2,2) + R(1,1,1,2)*X(1) *E             ))/X(1) 
 
                                          2                2  ( - P1(X(1)))                         2     2 
     RICCI(3,1):=RICCI(1,3):=( - SIN(X(2)) *R(1,1,1,3)*X(1) *E              - R(3,1,3,3))/(SIN(X(2)) *X(1) ) 
 
                                                              2  ( - Q1(X(1)))       2 
     RICCI(0,2):=RICCI(2,0):=( - (R(2,2,2,0) - R(0,2,0,0)*X(1) *E             ))/X(1) 
 
                                                              2  ( - P1(X(1)))       2 
     RICCI(1,2):=RICCI(2,1):=( - (R(2,2,2,1) + R(1,2,1,1)*X(1) *E             ))/X(1) 
 
                                                     3  ( - P1(X(1)))                         3  ( - P1(X(1))) 
     RICCI(2,2):=RICCI(2,2):=( - (2*R(2,2,2,2) + X(1) *E             *DF(P1(X(1)),X(1)) - X(1) *E             *DF(Q1(X 
 
                        2  ( - P1(X(1)))         2          2 
     (1)),X(1)) - 2*X(1) *E              + 2*X(1) ))/(2*X(1) ) 
 
                                          2                                    2     2 
     RICCI(3,2):=RICCI(2,3):=( - SIN(X(2)) *R(2,2,2,3) - R(3,2,3,3))/(SIN(X(2)) *X(1) ) 
 
                                       2                2  ( - Q1(X(1)))                         2     2 
     RICCI(0,3):=RICCI(3,0):=(SIN(X(2)) *R(0,3,0,0)*X(1) *E              - R(3,3,3,0))/(SIN(X(2)) *X(1) ) 
 
                                          2                2  ( - P1(X(1)))                         2     2 
     RICCI(1,3):=RICCI(3,1):=( - SIN(X(2)) *R(1,3,1,1)*X(1) *E              - R(3,3,3,1))/(SIN(X(2)) *X(1) ) 
 
                                          2                                    2     2 
     RICCI(2,3):=RICCI(3,2):=( - SIN(X(2)) *R(2,3,2,2) - R(3,3,3,2))/(SIN(X(2)) *X(1) ) 
 
                                          4     2        ( - P1(X(1)))                           ( - P1(X(1))) 
     RICCI(3,3):=RICCI(3,3):=( - SIN(X(2)) *X(1) *(X(1)*E             *DF(P1(X(1)),X(1)) - X(1)*E             *DF(Q1(X 
 
                     ( - P1(X(1)))                                  2     2 
     (1)),X(1)) - 2*E              + 2) - 2*R(3,3,3,3))/(2*SIN(X(2)) *X(1) ) 
 
 
     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); 
 
                  4                                  4  ( - 2*P1(X(1)))                    4  ( - 2*Q1(X(1)))       4 
     R:=(SIN(X(2)) *(2*R(2,2,2,2) + 2*R(1,1,1,1)*X(1) *E                + 2*R(0,0,0,0)*X(1) *E                + X(1) * 
 
          ( - P1(X(1)))                                           4  ( - P1(X(1)))                  2         4  ( - 
         E             *DF(P1(X(1)),X(1))*DF(Q1(X(1)),X(1)) - X(1) *E             *DF(Q1(X(1)),X(1))  - 2*X(1) *E 
 
         P1(X(1)))                                3  ( - P1(X(1)))                           3  ( - P1(X(1))) 
                  *DF(Q1(X(1)),X(1),X(1)) + 4*X(1) *E             *DF(P1(X(1)),X(1)) - 4*X(1) *E             *DF(Q1(X( 
 
                           2  ( - P1(X(1)))         2                              4     4 
         1)),X(1)) - 4*X(1) *E              + 4*X(1) ) + 2*R(3,3,3,3))/(2*SIN(X(2)) *X(1) ) 
 
 
     END OF RICCI TENSOR AND SCALAR CALCULATION; 
 
     LEAVING REDUCE ... 
     *** 

                                                                                
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;