ࡱ>  bd} 0bjbj55 @__Do, (5 UUUiii8M%\iS'O:e:{:{:V;T=L>PPPPPPPVXPU?V;V;??PUU{:{:>SZZZ?>_U{:U{:PZ?PZZzBeN{:гkoiܞN"FjsPTS0SGN@Y*0@YeNeN&@YUN??Z?????PPZ???S????@Y????????? : 1'GFE' '3*A'/G '2 F1E 'A2'1 REDUCE   AG13* 9FH'F ............................................................................................................................... 5A-G EB/EG ..................................................................................................................................... 2 "4F'JJ (' E-J7 F1E 'A2'1 REDUCE ................................................................................ 3 1.*91JA identifier ............................................................................................................... 7 2./3*H1G' ( commands ) .............................................................. 7 3.9ED1G' ( operators ) .............................................................. 19 3.1.9ED1G' 1J'6 ( arithmatic operators ) ................................ 23 3.2.9ED1G' (HD ( Boolean operators ) ................................................ 25 3.3.9ED1G' ,(1 ( algebraic operators ) ................................ 26 4.*H'(9 1J'6 .'5 ............................................................................................................... 31 Declarations.5 ............................................................................................................... 32 6.3H&J G' ( switches ) ................................................................................................ 37 7.E'*1J3 G' ................................................................................................................ 42 8.J F*G ............................................................................................................................ 51 EB/EG F1E 'A2'1 REDUCE J '(2'1 E-'3('* ,(1J /1 1J'6J, AJ2J H EGF/3J '3*. *HDJ/ 'JF F1E 'A2'1 -'5D *D'4 E4*1 'A1'/ (3J'1J '3* HDJ EF4# '5DJ 4D J1J "F '1 "F*HFJ G1F /1 1963 '3*. REDUCE (1'J 'HDJF ('1 /1 GEJF 3'D *H2J9 4/ H '2 'JF 2E'F (G (9/ 5/G' FA1 '2 1'G G'J E.*DA /1 *H39G J 'JF F1E 'A2'1 /.JD (H/G 'F/. (1'J E+'D, ,'F AJ, G1(1* EPDPF, HJF A1J/ FHF, "1*H1 FH1EF H 'P(P1G'1/ 41HA1 '2 'A1'/J G3*F/ G /1 4D J1J G3*G J REDUCE H (3*G G' (package ) J E1*(7 41* E/'HE /'4*G 'F/. (1.J B'(DJ* G'J F1E 'A2'1 REDUCE 9('1*F/ '2: E1*( 3'2J H (37 *H'(9 F/,EDG 'J H *H'(9 HJ' 'F,'E 'FH'9 ,'0'1J G' 3'/G 3'2J (G 5H1* '*HE'*J J' *-* F*1D '1(1 'F,'E E-'3('* E'*1J3J *9JJF /B* /D.H'G /1 E-'3(G J '9/'/ 5-J- J' -BJBJ 'E'F *91JA *H'(9 ,/J/ H *H39G J syntax (1F'EG E4*B J1J H 'F*1'D J1J *-DJDJ A'*H1J1J '2 F/,EDG 'J G' -D E9'/D'* ,(1J E.*DA 'E'F F*1D 4D .1H,J (1F'EG 'F,'E E-'3('* E1(H7 (G 'FH'9 *H'(9 .'5 ( (3D, D'F/1, 'E', 2*' H ... ) 'F,'E E-'3('* E1(H7 (G E'*1J3 G'J /J1', B'(D '3*A'/G (1'J E-'3('* AJ2J 'F1J G'J ('D' Reduce  F1E 'A2'1 (1' E-'3(G  9ED1G' ,(1 (' /B* ('D'3*. (/HF *H,G (G 'F G B/1 'F E-'3(G ~/G ('4/. 'F F1E 'A2'1 *H'F' '1 (' F/ ,EDG ' G' (G 5H1* G' E.*DA 1' /'1/ H GEFF *H'F' A'*H11 H GEF7H1 ~.4 "FG'. (G 9D'HG Reduce B'/1 (G 'F*1'D 1 H E4*B 1A*F '3*. '1 1/F (' "1'G G'  A1'F/G' H 9ED1G' G '(2'1 '1 A2 'F1 G' ('D' '3* F2 ,2! E('F 'F F1E 'A2'1 '3*. (1' 1'-* '1 'E'F *4D A'D H '3*A'/G '2 "F (1' 0.1G  *1'1 H ' FG /'1 F*', (1F'EG /1 'F F1E 'A2'1 *9(G 4/G '3*.  (1*1 Reduce E-'H1G ' (H/F "F '3*. (/F 5H1* G '1(1 E *H'F/  ,EDG  ,(1 1' H'1/ F/  F*,G '14 1' E4'G/G F/ H 3~3 (G '/'EG  (1F'EG (~1/'2/. (1' '3*A'/G '2 F1E 'A2'1 (G 5H1* E-'H1G ' ('D'.5 (1' E-'3('* 7HD'F  Reduce E *H'F/ /1 -'D* Batch mode '3*A'/G E 4H/ G /1 A2 (3'1 ~1'1(1/ '3*. /1 'F -'D* G1 /3*H1 G FH4*G E 4H/ H (1F'EG ~4 E1H/ (G 7H1 GE2E'F F*', '1 //G E 4H/ H '1(1 1'-* *1 (G '/'EG '1 E ~1/'2/. 41H9 (1F'EG FH3: 2E'F G F1 E 'A2'1 Reduce *H37 3'E'FG A1'.H'F/G 4H/   ~:'E (G 5H1* 21 FH4*G E 4H/: Reduce 3.8 , 15-July-2003 & G *'1. H E/D "F G1 ('1 E *H'F/ *:1 '(/. -'D 4E' E *H'F/ G ,EDG 1' /1 F1E 'A2'1 (FH3/. /1 Reduce G1 ,EDG ('  Semicolon *E'E E 4H/ G FE'41 'F,'E 'F ,EDG H ":'2 /3*H1 (9/ '3* E'FF/: )x+y+z)^2; 'F /3*H1 (G 7H1 E9EHD ('  '1'*1 /1 '/'EG E '(/ (/EG Return /1 5A-G D/G' ASCII) *' F1E 'A2'1 1' '57D'-'K (/'1 F/ G 'F '1 ,EDG 1' /'1/ E F/  E-'3(G E F/ H F*,G 1' (G 5H1* 21 E FH3/ X2+2*X*Y+2*X*Z+Y2+2*Y*Z+Z2 -'D 'F E+'D 3'/G 1' (G 7H1 3*1/G *1 (113 E FE G 1',9 (G 1H4 G Reduce '1 E F/ (4*1 '/ (1E. '(*/'  ('/ E*01 4HE G Reduce E'FF/ (BG  2('FG' 'E~H*1 (' E*:1G' H +'(* G' 31H'1 /'1/. HD /1 E-'3(G  9('1* B(D  E*:1 E *H'F/ E3*BD'K (113 4H/ 9F '2 71B F'E .H/4 *91A 4H/ G /1 2('F G' /1 (1F'EG FH3 'F 'E'F H,H/ F/'1/. E-'3('* ,EDG G' ,(1 '2 BH'FF ,(1 /(13*'F ~1H E F/ H *FG' 2 :19'/ /1 E+'D ('D' 'F (H/G G 9('1* 3*14 /'/G 4/G '3*. Reduce GEH'1G 9('1* G' 1' *' ,' G 'E'F H'1/ 3*14 E /G/  -3( E*:1G' E.*DA ,/' E F/ H (1'3'3 1H4 .'5 (G *1*( /1,G  ,ED'* E1*( E F/. 'D(*G 3*1/  /1,G (F/ E*:1G'  FH9 .1H, H :1G *-* F*1D '1(1 '3*. F*G  /1 1',9 (G FH9 '1'*1G' H1H/ H .1H, '3*. /1H'B9 H1H/ E *H'F/ (G G1 /H 5H1* -1HA H H (21 ('4/ H '1'*1G' .1H, GEH'1G (' -1HA H 'F/. /1 FG'* '2  '1'*1 9// '3*A'/G E 4H/ *' (*H'F F*,G 1' /1 E-'3('* (9/ '1,'9 /'/. (1' FE'4 (4*1 H G' F1E 'A2'1 E+'D 21 1' (113 E FE: For i:=1:40 product I; /1 'F E+'D F*,G EB/'1 !40 .H'G/ (H/ G 9// 21 '3* : 815915283247897734345611269596115894272000000000 4E' E *H'F/ F*,G E4'(G 1' (' FH4*F 'F ,EDG (/3* "H1/. Factorial 40; HF /1 E-'3(G  ,(1 ,H'( /BB EH1/F81 '3*  'ED'K 61H1 '3* G '2 '9/'/ 5-- (' /B* E4.5 '3*A'/G 4H/. E'FF/ E+'D ('D' (G 9D'HG 9('1* FOR /1 ,EDG  ('D'  '2 -DBG G' B'(D '3*A'/G '3* G Reduce (1' 1'-* '1(1 '3*A'/G '2 "F 1' EEF 3'.*G '3*.  /1 '2 AH'/ Reduce B'(D* *9F -/'+1 1BE G'  9// '3*. /1 -'D* 9'/ 100 'EF 1BE G1 9// 5-- "A1F 1BE B'(D '3*A'/G /1 Reduce '3*. /1 (96 EH'1/ D'2E '3* '2 F*,G   9('1* ,(1 /1 '/'EG  E-'3('* '3*A'/G 4H/  1H4 (1' 'F *91A  E*:1 ,// (' 9D'E* =: '3* E'FF/ U:=(x+y+z)^2 /1 'F -'D* E *H'FE '2 V (G ,' 9('1* ('D' /1 '/'EG  (1F'EG '3*A'/G FE. GEFF F*,G  ".1F 9('1* ,(1 /1 (1F'EG GEH'1G (' F'E WS (Workspace) /1 Reduce 0.1G E 4H/ H E *H'F/ EH1/ '3*A'/G B1'1 1/. (G 9FH'F E+'D 9('1*: df(ws,x); '1 (D'A'5DG ~3 '2 9('1* B(D ('/ E4*B 9('1*  1' F3(* (G x E-'3(G E F/ (G 7H1 E4'(G: Int(ws,y); 'F*1'D G' 9('1* 1' F3(* (G y -3'( E F/. Reduce GEFF B'/1 (G '1 (' FE'/ E'*13 '3*. (G 9FH'F E+'D: matrix m(2,2); M 1' (G 9FH'F  E'*13 2(2 (1' (1F'EG *91A E F/ H m:= mat( (a,b), (c,d) ) ; EB'/1 E'*13 m 1' *9F E F/. -'D 9('1'* G 4'ED 9ED'* G' ,(1 (1 1H m E 4HF/ 1' E *H'F E4'G/G 1/ E+D'K m/1 G E'*13 E9H3 1' E /G/ H  G -'5D4  E'*13 /1 '3* H /1 FG'* det(m) G E-'3(G FF/G /*1EF'F E'*13 m '3*. Reduce 4'ED E-/H/G 3*1/G ' '2 9ED1G' ,'F4F '3*. 'F F1E 'A2'1 *H'(9 EB/E'* 1' E 4F'3/ HD 1H'(7 -'E (1 "FG' 1' (G 7H1 D /1 (1 FE 1/. (G 9FH'F E+'D E,EH9 *H'(9 E+D+'* (G 7H1 .H/(G .H/ (G -'5D 61( 9('1'* 2'HG ' *(/D FE 4H/ HD '1 '1(1 (.H'G/ E *H'F/ (G 9FH'F E+'D *91A F/: ( sin(a+b) + cos(a+b))*(sin(a-b)  cos(a-b)) Where 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; (G 7H1G 9D'E* *D1H' B(D '2 E*:1G' x H y FE''F1 9FH'F GEG  EB'/1 'F E*:1G'3*. F*,G  E-'3('* (G 5H1* 21 E-'3(G E 4H/. -(cos(2*A)+sin(2*B))  B'(D* (3'1 ~1'3*A'/G (1F'EG H GEF7H1  '2 F/ 1H4 /'/G G' .1H, Reduce  B'(D* FE'4 .1H, G' /1 A1E FORTRAN '3*. 'F .1H, G' (9/'K E *H'FF/ /1 A1E E-'3('* 9// (1E(F' FORTRAN '3*A'/G 4HF/. 'F .'5* ('D'.5 (1' A1EHD G' ,(1 G D'2E '3* (G 9FH'F E('F E-'3('* 9// '3*A'/G 4HF/ '1(1/ /'1/. (G 9FH'F E+'D  9('1* On fort; df (log(x)* (sin(x) + cos(x))/ sqrt(x),x,2) F*,G 21 1' /1 (1 .H'G/ /'4*: ANS=(-4.*LOG(X)*COS(X)*X**2-4.*LOG(X)*COS(X)*X+3.* . LOG(X)*COS(X)-4.*LOG(X)*SIN(X)*X**2+4.*LOG(X)* . SIN(X)*X+3.*LOG(X)*SIN(X)+8.*COS(X)*X-8.*COS(X)-8. . *SIN(X)*X-8.*SIN(X))/(4.*SQRT(X)*X**2) 'F '1(1/G' ,(1 FE''F1 G1G  ,(1 Reduce '3*. Reduce (1'3'3 2('F '3*'F/'1/ (1F'EG 12 4/G '3*.  -'D* FE'/F F2 (1' '1 (' ,ED'* 2('F H,H/ /'1/. 'F ,EDG 1'E1 2('F Lisp 1' (G /F('D .H'G/ /'4* Symbolic car' (a) ; ,'(,' (F G1 /H -'D* 2('F 'E'F ~01 '3*. -'D (G E91A * * /3*H1G' E ~1/'2E: 1.*91JA identifier (1' F'E E*:J1G', (13( 9('1* /3*H1 go to , F'E "1'JG G', E'*1J3 G', 9ED1G' H procedure G' '2 identifier '3*A'/G E FJE G E *H'F/ 4'ED -1HA H '9/'/ ('4/ HD 'HDJF -1A "F F('J/ 9// ('4/. (J4*1JF *9/'/ -1HA E,'2 (1' J identifier /1 3J3*E G' E.*DA E*A'H* '3* HD E9EHD'K -/H/ 100 -1A '3*. G1F/ (G*1 '3* (1' '~ 1'-* *1 *9/'/ -1HA E*1 '2 25 'F*.'( 4H/. 2./3*H1G' (Commands) Syntax: ; and $ 'JF /H -/ 2'1G G' 1' *9JJF EJ FF/. G ; F*J,G 1' '~ EJ F/ HDJ $ F*J,G 1' '~ FEJ F/. /'.D J 1HG  HYPERLINK \l "group" <<...>> J'  HYPERLINK \l "block" block G1 /H (1'J ,/' 1/F 2'1G G' (G '1 EJ 1HF/ H A1BJ (' GE F/'1F/. HF '2 /'.D block (/HF H,H/ /3*H1  HYPERLINK \l "return" return J2J '~ FEJ 4H/. % or comment (1'J FH4*F *H6J- /1 /'.D (1F'EG '3*A'/G EJ 4HF/. %: G1 J2J (9/ '2 % *' 'F*G'J .7 ',1' FEJ 4H/ '2 ,EDG ; H $ Comment: '2 DEG J comment *' 'HDJF ; H J' $ ',1' FEJ 4H/. comment (1'J 0'4*F *H6J- F/ .7J EF'3( '3* : 1: df (x**3+y,x)% this is a comment; 1: ; 3x2 ; .7 'HD ',1' F4/G '3*. 1: x:a** comment-a is positive 1: a is an integer ;; X:=a2 'HDJF; (1'J ~'J'F comment '3* H /HEJ (1'J '~ F*J,G. <<...>> group /1 ,'JJ G REDUCE 'F*8'1 J 2'1G 1' /'1/  F/ 2'1G 1' (' GE J 1HG EJ F/. E+D'K /1  HYPERLINK \l "if" if& then J'  HYPERLINK \l "repeat" repeat& until J'  HYPERLINK \l "while" while& do /'.D -DBG J' 9('1* 417J ('J/ J 2'1G /'4*G ('4JE. J' B(D '2 do G J 2'1G D'2E /'1JE. 2'1G G'J /'.D 1HG (' ; J' $ ,/' EJ 4HF/ H A1BJ (' GE F/'1F/. 1HG EB/'1 ".1JF 2'1G 1' (1 EJ 1/'F/. ~3 '1 (JF ".1JF 2'1G H (1'* (3*G (<<) ; J' $ B1'1 (J1/  EB/'1 5A1 J' nill (".1JF 2'1G) (11/'F/G EJ 4H/. ~3 /1 -'D* DJ (9/ '2 ".1JF 2'1G ; J' $ FEJ 0'1JE. begin & end (block) J block EJ *H'F/ 9EDJ'*J 1' 'F,'E /G/ E+D 1: begin for i:=1:3 do write I end; 1 2 3 HDJ (1 .D'A  HYPERLINK \l "group" group *' HB*J /3*H1  HYPERLINK \l "return" return /'.D "F F('4/ EB/'1J (1FEJ 1/'F/. /3*H1  HYPERLINK \l "return" return ('J/ ".1JF /3*H1 B(D '2 end ('4/. 1: begin scalar n; n:=1; b:=for i:=1:4 product (x-i); return n end; 1 2: b; X4-10x3+35x2-50x+24 E*:J1G'J E-DJ (local) /1 /'.D  HYPERLINK \l "block" block (D'A'5DG (9/ '2 begin ('J/ *91JA 4HF/ H EJ *H'FF/ scalar real J' integer ('4F/. HDJ "1'JG 'J G /'.D  HYPERLINK \l "block" block *91JA EJ 4H/  /1 G1 -'D* global '3*. /3*H1  HYPERLINK \l "let" let GE /1  HYPERLINK \l "block" block *#+J1 global /'1/. B(D '2 end  ; J' $ D'2E FJ3* HDJ (9/ '2 "F ; J' $ J' (1'* (3*G 'J E*F'81 (' (1'*J G B(D'K ('2 4/G  ('J/ ('4/. Goto or go to <label>: <statement> go to <labeled-statement>; /3*H1 goto AB7 /'.D J  HYPERLINK \l "block" block '3*A'/G EJ 4H/ H (1F'EG (G .7J G (13( (label)EH1/ F81 1' /'1/ EJ 1H/. (G*1 '3* (G 4D go to '3*A'/G 4H/. /'.D  HYPERLINK \l "block" block AB7 2'1G G'J ('D'*1JF 37- EJ *H'FF/ (13( 0'1J 4HF/ FG 2'1G G'J E+D'K /'.D  HYPERLINK \l "group" <<...>> J'  HYPERLINK \l "while" while& do H ... G /1  HYPERLINK \l "block" block H,H/ /'1F/. for for <var>:= <number> step <number> until <number> <action> <exprn> for <var>:= <number> : <number> <action> <exprn> for each <var> in <list> <action> <exprn> <var> EJ *H'F/ G1 E*:J1J (G :J1 '2 t H nil ('4/. <action> EJ *H'F/ 9EDJ'*J E'FF/ sum, product, do join, collect, H ... ('4/ H (G /F('D "FG' ('J/ J 2'1G J *FG' J'  HYPERLINK \l "group" group J'  HYPERLINK \l "block" block (J'J/. F/ E+'D: 1: for i:=1:10 sum i ; 55 1: for a:=-2 step 3 until 6 product a; -8 1: m:=0$ 2: for s:=10 step -1 until 3 do << d:=10*s; m:=m+d>>; 3:m; 520 1: for i:=1:4 collect 1/i ;  EMBED Equation.3  1: for each x in {q,r,s} sum x**2; q2+r2+s2 '1 -/ ('D' '2 -/ ~'JJF E*1 ('4/ GJ 9EDJ'*J 'F,'E FEJ 4H/. E*:J1 *1'1 FF/G (& ,s,a,i) /1 2'1G J local , for '3* H EB/'1 J E*:J1 (' GE'F F'E 1' *:JJ1 FEJ /G/ EJ *H'F F/ 2'1G J for 1' (G 5H1* *H /1 *H '3*A'/G 1/ (G 417J G J2J G 2'1G G'J /'.DJ (1EJ 1/'FF/ (1'J 2'1G G'J .'1,J E9FJ /'1 ('4/. if if <condition> then <statement> else <statement> <condition> ('J/ J 2'1G EF7BJ J' *1J(J '2 "FG' ((G H3JDG J or , and) ('4/. 2'1G J EF7BJ: EB/'1 Boolean (1EJ 1/'F/ J9FJ '1 5A1 J' nil ('4/ false (1EJ 1/'F/ H '1 G1 EB/'1 /J1J ('4/  true (1EJ 1/'F/. <statement> ('J/ J 2'1G J *FG' J' J  HYPERLINK \l "group" group J'  HYPERLINK \l "block" block ('4/. F/ E+'D: 1: x:=9$ 2: if number p x and x<20 then y:=sqrt (x) else write "illegal"; 3 1: x: =12$ 2: a:=if x<5 then 100 else 150; a:=150 3: b:=u**(if x<10 then 2); b:1 4: bb:=u**(if x>10 then 2); bb:=u2 '1 else F/'4*G ('4JE H 417 if (1B1'1 F4H/  EB/'1 2'1G 5A1 .H'G/ (H/ E+D .7 3 /1 E+'D /HE '1 (G ,'J <undition> 2'1G J :J1 417J (0'1JE  E+D J +'(*  true /1 F81 1A*G EJ 4H/. 'JF /3*H1 *H /1 *H GE EJ *H'F/ '3*A'/G 4H/: If <condition> then if <condition> then <action> else <action> If <condition> then <action> else if <condition> then <action> else <action> Repeat Repeat <statement> until <condition> <statement> ('J/ J 2'1G J *FG' J' J  HYPERLINK \l "group" group J'  HYPERLINK \l "block" block ('4/. <condition> ('J/ J 2'1G J EF7BJ J' *1J(J '2 "FG' ((G H3JDG J  HYPERLINK \l "andor" and H HYPERLINK \l "andor"  or) ('4/. <statement> -/'BD J ('1 ',1' EJ 4H/. EH'8( loop (J FG'J* ('4J/! 1: m:=3$ 2: repeat <<write 100*x*m; m:=m-1>> until m=0; 300x 200x 100x 1: m:=-1$ 2:repeat <<write m; m:=m+1>> til m>=0; -1 Return 'JF /3*H1 J EB/'1 1' '2 /'.D  HYPERLINK \l "block" block (1EJ 1/'F/. begin <statement> return <expression> end; EB/'1 <experession> (11/'F/G EJ 4H/. F/ E+'D: 1: begin write  yes ; return a end; yes a 1: procedure dump(a); 1: begin if number(a) then return a else return 10 end; Dump 2: dump(-5); -5 3: dump(x); 10 Return (/1 5H1* H,H/ ) GEJ4G ".1JF 2'1G B(D '2 .'1, 4/F '2  HYPERLINK \l "block" block '3*. (/HF return GJ EB/'1 (11/'F/G FE 4H/ HD 9EDJ'* block 'F,'E E 4H/. '1 return /1 /'.D  HYPERLINK \l "group" group J'  HYPERLINK \l "if" if& then& else '3*A'/G 4H/, 'JFG' ('J/ /'.D block ('4F/H (1'  HYPERLINK \l "for" for H  HYPERLINK \l "repeat" repeat& until H  HYPERLINK \l "while" while& do FE *H'F '2 return '3*A'/G 1/ -* '1 /'.D block ('4F/. /1 block G' *H /1 *H , return /1 block /'.D, EB/'1 1' (G block (9/ (1 E 1/'F/ FG (G .'1, '2 GEG  block G'. Procedure '2 'JF /3*H1 (1' *91JA 1/F J 9ED1 1J'6 (G 9FH'F J *'(9 (' E*:J1G' "F '3*A'/G E 4H/. Procedure <identifier>(<arg>,<arg>,& ); <body> <body> ('J/ J 2'1G  *FG' J' J  HYPERLINK \l "group" group J'  HYPERLINK \l "block" block ('4/. 1: procedure fac(n); 1: if not (fixp (n) and n>=0) 1: then rederr "choose non neg-integer only" 1: else for i:=o:n-1 productu i+1 ; Fac 2: fac(0); 1 3: fac(-5); Choose nonneg-integer only HB*J REDUCE  procedure 1' *,2JG 1/G H (' EHABJ* (G A1E B'(D '3*A'/G .H/ /1 "H1/ '3E "F 1' '~ EJ F/. /'.D *91JA J procedure FEJ *H'F J procedure /J1 *91JA 1/ HDJ EJ *H'F J procedure /J1 J' .H/ procedure 1' (G 4D ('24*J 5/' 1/. E*:J1G'J procedure 5H1J G3*F/ J9FJ E+D'K E*:J1 x 1(7J (G x 'J G /1 (1F'EG *91JA 4/G F/'1/. H EB/'1 "F 1' *:JJ1 FEJ /G/. (1'J E*:J1J G procedure 1' 5/' EJ F/  '1 /'.D procedure J2J (G "F assign 4/G ('4/ (:=) EB/'14 AB7 /'.D procedure *:JJ1 EJ F/ HDJ  HYPERLINK \l "let" let EB/'1 "F (G 5H1* global *:JJ1 EJ /G/. /3*H1 clear GE (1'J E*:J1G'J 5H1J *#+J1 global /'1F/. '~1'*H1 H "1'JG EJ *H'FF/ E*:J1 procedure ('4F/ H /1 G1 -'D EB/'14'F (G 5H1* global *:JJ1 EJ F/ HDJ E'*1J3 FEJ *H'F/ 1HJ procedure ('4/. G1 ~'1'E*1J G 5H1J F('4/ (5/' 2FF/G J procedure F('4/) H local GE F('4/ (G (D'A'5DG (9/ '2 begin /1 block *91JA 4H/) global /1 F81 1A*G EJ 4H/. '1 B1'1 ('4/ procedure J EB/'1 1' (11/'F/ /3*H1  HYPERLINK \l "return" return ('J/ ".1JF 2'1G B(D '2 .1H, '2 procedure ('4/. write write <item> , <item> , & . 1: write a," ", sin(x), "this is a string" ; A sin(x)this is a string 1: array m(10); 2: for i:=1:3 do write m(i):=2*i ; m(1)=2 m(2)=4 m(3)=6 3: m(3); 6 Item G'J J /3*H1 write GEG /1 J .7 '~ EJ 4HF E1 'JFG .JDJ 7HD'FJ ('4/. define '1(1/ /3*H1 define /1 E+'D 2J1 H'6- '3*: 1: define is = :=, xx=y+z ; 2: a is 10 ; a :=10 3: xx**2 ; y2+2yz+z2 4: xx :=10 ; y+z :=10 /1 (G '1 (1/F define ('J/ E1'B( (H/ HF (G 1'-*J FEJ *H'F "F 1' (/HF (3*F (1F'EG (G -'D 'HD (11/'F/. on and off off (1'J .'EH4 1/F J  HYPERLINK \l "switch" switch H on (1'J (G '1 'F/'.*F J  HYPERLINK \l "switch" switch '3*A'/G EJ 4H/: on switch-name; off switch-name; .'EH4 H 1H4F 1/F switch G' /1 toolbar B3E* switch GE 'E'F ~0J1 '3*. while while <condition> do <statement> <condition> ('J/ J 2'1G J EF7BJ J' *1J(J '2 "FG' ((G H3JDG J and H or) ('4/. <statement> ('J/ J 2'1G J *FG' J' J group J' block ('4/. 1: a:=10 ; 2: while a<=12 do << write a; a:=a+1>> ; 10 11 12 let (1'J *91JA BH'FJF ,'J2JFJ Let <identifier> = <expression> <identifier> EJ *H'F/ G1 J2J (G :J1 '2 "1'JG ('4/ '2 ,EDG J <expression> 1: let a=sin(x); 2: b: a; b:sin(x) 3: exp(a) ; esin(x) A1B let (' '~1'*H1  HYPERLINK \l "assign" := 'JF '3* G := EB/'1 9('1* 3E* ~ .H/ 1' fix EJ F/ HDJ (' let  E*:J1 *:JJ1'* (9/J 1' /F('D EJ F/: 4: let c=a ; 5: a:= xn2 $ 6: exp(a) ;  EMBED Equation.3  7: exp(b) ; esin(x) 8: exp(c) ;  EMBED Equation.3  EJ (JFJE G c *:JJ1 a 1' /F('D 1/G '3* HDJ b G '(*/' EB/'1 a (G "F assign 4/G (H/  GE'F EB/'1 'HD 1' -A8 1/G '3*. 1: operator h; 1:array q(10); 2: let h(u,v)=u-v; 2:let q(1)=15; 3: h(u,v); **** substitution for o not allowed u-v 4: h(x,y); h(x,y) HB*J /1 *91JA 9ED1/ '~1'*H1 '2 let '3*A'/G EJ FJE  'JF 9ED1/ AB7 (1'J E*:J1G'J /BJB'K (G GE'F F'EJ G '3*A'/G 4/G E9*(1 '3*. "1'JG (G GF'E *91JA (' 5A1 EB/'1/GJ 4/G  (F'(1'JF let q(1)=15; EF,1 (G ~J:'E .7' EJ 4H/.(let o=15;) HDJ '1 '96'J "1'JG J' /1'JG G'J G'J E'*1J3 E-*HJ J E*:J1 J' 9('1* E9*(1 (1'J let ('4F/ *#+J1 let (G 5H1* global 1HJ "F E*:J1 H 9('1* '9E'D EJ 4H/  J9FJ FG AB7 1HJ E*:J1 /'.D E'*1J3 J' "1'JG (DG E*:J1 /1 GEG ,'J (1F'EG. (1'J ,DHJ1J '2 'JF 9H'16 (G*1 '3* let (H GEJF 7H1  HYPERLINK \l "clear" clear) (1'J "1'JG H E'*1J3 '3*A'/G F4H/ H (G ,'J "F '2  HYPERLINK \l "assign" := '3*A'/G 4H/. AB7 *G * ',2'J "1'JG EJ *H'FF/ (1'J let 3E* ~ *3'HJ B1'1 (J1F/. HDJ E'*1J3 EJ *H'F/ (G 9FH'F J D 3E* ~ *3'HJ ('4/ (G 417J G 3E* 1'3* GE E'*1J3 ('4/. E*:J1G'J local (G /'.D  HYPERLINK \l "block" block *91JA EJ 4HF/) FEJ *H'FF/ 3E* ~ *3'HJ let B1'1 (J1F/ HDJ D block EJ *H'F/ 3E* 1'3* *3'HJ let ('4/ (' 'JF syntax: for all (vars) let <operator> (<vars>) = <block> let (1'J active 1/F J rule list GE (G '1 EJ 1H/: 1: trigl :={cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2 , Cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2}$ 2: let trigl ; 3: cos(a)*cos(b) ;  EMBED Equation.3  F/ F*G /1 '3*A'/G '2 let: A*B : let A*B=C (1 1' E3'HJ C B1'1 EJ /G/. A : let A+C=C (2 1' E3'HJ C-B B1'1 EJ /G/. A : let A-B=C (3 1' E3'HJ C+B B1'1 EJ /G/. A : let A/B=C (4 1' E3'HJ B*C B1'1 EJ /G/. clear (1'J -0A 1/F *3'HJ G' J' BH'FJF ,'J2JFJ Clear <identifier> <identifier> EJ *H'F/ '3'D1   HYPERLINK \l "matrix" E'*1J3   HYPERLINK \l "array" "1'JG J' F'E  HYPERLINK \l "procedure" procedure ('4/. 1: array a(2,3) ; 2: a(2,2) :=15$ 3: clear a; 4: a(2,2); Declare A operator? (Y or N) 5: let x=y+z; 6: sin(x); Sin(y+z) 7: clear x; 8: sin(x); Sin(x) 9: let x^5=7; 10: clear x; 11: x^5; 7 12: clear x^5; 13: x^5;  EMBED Equation.3  '1 clear (1' ',2' HYPERLINK \l "array"  "1'JG '3*A'/G 4H/, /1 5H1* G 'JF ,2! E-*H 9('1* :J1 E9*(1 (1' clear ('4/ ( E+D 9// +'(* ) ~J:'E error E /G/H /1 :J1 'JF 5H1* E*:J1 EH,H/ /1 'JF ,2! ~' E 4H/. (1' E'*1J3 G' '1 9('1* :J1 E9*(1 ('4/ error 8'G1 E 4H/ H /1 :J1 'JF 5H1* GJ 9ED 'F,'E FE 4H/. (1' 'JFG ',2' "1'JG J' E'*1J3 1' E3'H 5A1 B1'1 /GJ/ '2 clear '3*A'/G FFJ/. 'JF '1 F*J,G  EH1/ 'F*8'1 4E' 1' F.H'G/ /'4*. HF ',2' FG E-*H 5A1 .H'G/ (H/ H FG E-*H GJ J2 /J1. GE'F 7H1 G /1 E+'D ".1 E (JFJ/ HB* J 9('1* ( *H'F, 61( H ... ) (' J2 ,'J2JF 4/G, (1' ~' 1/F 'JF ,'J2JF D 9('1* *' B(D '2 = ('J/ /1 "1HE'F clear FH4*G 4H/. Clearrules (1' ~' 1/F J rule list '2 clearrules '3*A'/G E FJE: 1:trig 1: ={cos(~x)cos(~y) => (cos(x+y)+cos(x-y))/2; cos(~x)sin(~y) => (sin(x+y)-sin(x-y)/2}$ 2: let trigl ; 3: cos(a)*cos(b);  EMBED Equation.3  4: clearrules trigl ; 5: cos(a)*cos(b); cos(a)cos(b) for all for all <identifier> let <let statement> J': for all < identifier> such that <condition> let <let statement> 1: for all x let f(x)=sin(xn2); Declare f operator? (Y or N) 2: Y 3: f(a); Sin (a2) 4: operator pos; 5: for all x such that x>=0 let pos(x) = sqrt(x+1) ; 6: pos (5);  EMBED Equation.3  7: pos(-5); Pos(-5) 8: clear pos; 9: pos(5); declare pos operator? (Y or N) (1'J ~' 1/F ,'J2JFJ *H37 for all (G 'JF E+'D *H,G FJ/: 1: for all a such that a>o let xna =1; 2: xn4; 1 3: clear xna ; *** xna Not Found 4: for all a clear xna; 5: xn4; 1 6: for all a such that a>0 clear xna ; 7: xn4; x4 for all & let E9'/D *91JA J  HYPERLINK \l "procedure" procedure '3* (' 'JF *A'H* G /1 procedure EB/'1 ~'1'E*1 5H1J (5/' 2FF/G J procedure) (G 5H1* global 9H6 FEJ 4H/ -*J '1 /'.D procedure EB/'1J (G "F assigne 4/G ('4/ (/1 'JF 1'(7G "1'JG G' '3*+F' G3*F/) /3*H1G'J /J1J GE *-* 9FH'F general commands /'1JE G *H6J- "FG' (G '.*5'1 FJF '3*: bye (1  EMBED Equation.3  '2 J 37- (1F'EG .'1, EJ 4H/ H (G 37- (level) (9/J EJ 1H/ H '1 /1 ('D'*1JF 37- ('4JE '2 REDUCE .'1, EJ 4H/. 'JF /3*H1 E9'/D quit '3* display(<n>) (2  EMBED Equation.3  <n> .7 B(D 1' '~ EJ F/ H '1 ~1'F*2 .'J ('4/ J' <n> '2 *9/'/ .7H7 (J4*1 ('4/ GEG J .7H7 1' '2 'HD '~ EJ F/. rederr <message> (3  EMBED Equation.3  J ~J:'E error '2 /'.D  HYPERLINK \l "procedure" procedure J'  HYPERLINK \l "block" block '~ EJ F/. retry (4  EMBED Equation.3  ".1JF 2'1G 'J G EF,1 (G error 4/G '3* 1' /H('1G 'E*-'F EJ F/. load-package "<package-name>" (5  EMBED Equation.3  package EH1/ F81 1' load EJ F/ H /1 B3E* toolbar (1F'EG GE B'(D /3*13J '3*. 3.9ED1G' (Operators) Syntax Assign := EB/'1 3E* 1'3* 1' (G 3E* ~ assign EJ F/. : = '2 3E* 1'3* '9E'D EJ 4H/: 1: a:=b:=c $ 2: a; c 3: b; c GEG J 9('1*G' /1 'JF 2F,J1G (G :J1 '2 9('1* ".1 (1'(1 (' c B1'1 /'/G EJ 4HF/. 3E* ~ EEF '3* J expression ('4/: 1: y+b:=c & 2: y; -(b-c) : = (1'J ',2'J HYPERLINK \l "array"  "1'JG G' '1(1/ /'1/ HDJ (1'J D "1'JG FG. HDJ (1'J /1'JG G'J HYPERLINK \l "matrix"  E'*1J3 J' D J E'*1J3 EJ *H'F '2 "F '3*A'/G 1/. '3*A'/G '2 3'.*'1 ('24*J E+D a:a+b GE E'F9J F/'1/. equal sign = /1 2'1G G'J 417J ( E+D  HYPERLINK \l "if" if& then& else J'  HYPERLINK \l "repeat" repeat& until ) H /1 *91JA  HYPERLINK \l "equation" equation (G '1 EJ 1H/. 1: a:=4 $ 2: if a=10 then write "yes" lese write "no"; No + - * / '1(1/ + H  H'6- '3*. '3*A'/G '2 "FG' (1'J  HYPERLINK \l "matrix" E'*1J3 G' H  HYPERLINK \l "equation" equation G' GE EEF '3*. '~1'*H1 * EJ *H'F/ (1'J 61( E'*1J3G' (' (9/ EF'3( GE '3*A'/G 4H/. /1 9('1*J E+D REDUCE ,2x (G 7H1 6EFJ '~1'*H1 * 1' /1 F81 EJ J1/ HF  HYPERLINK \l "identifier" identifier FEJ *H'F/ (' 9// 41H9 4H/. '~1'*H1 / GE (1'J E'*1J3 G'J E1(9J EJ *H'F/ '3*A'/G 4H/: A/B=A*B^-1 (G 417J G B H'1HF ~0J1 ('4/. 1/A GE E9H3 A 1' EJ /G/. 1: 100/6 ;  EMBED Equation.3  2: 16/2/x ;  EMBED Equation.3  3: on rounded ; 4: 55/4; 8.75 ** or ^ 1: x**y**z; xyz 2: x**(Y**Z);  EMBED Equation.3  3: on rounded; 4: 2**pi ; 8-82497782708 E'*1J3 E1(9J EJ *H'F/ (G *H'F E+(* J' EFAJ ((G 417 E9H3 ~0J1J) (13/. 9('1* G'J '3'D1 H  HYPERLINK \l "equation" equation G' EJ *H'FF/ (G *H'F 31J H '94'1J (13F/. <= < > = > 'JF '~1'*H1G' AB7 (1'J EB'J3G (JF E*:J1G'JJ G E*F'81 (' J 9// G3*F/ '3*A'/G EJ 4HF/ H /1 5H1* (1B1'1J 417  true (1EJ 1/'FF/. true 'J G (1EJ 1/'FF/ AB7 (G 9FH'F 9('1'* 417J /1  HYPERLINK \l "while" while& do J'  HYPERLINK \l "repeat" repeat& until J'  HYPERLINK \l "if" if& then& else '3*A'/G EJ 4H/. ~ '~1'*H1 (tild) ~ J free variable 1' F4'F EJ /G/. and & or and : '1 G1 /H argument /13* ('4/  true (1EJ 1/'F/. or : '1 -/'BD JJ '2 argument G' /13* ('4/  true (1EJ 1/'F/. AB7 /1 9('1*G'J 417J '3*A'/G EJ 4HF/. '2 3E* ~ '9E'D EJ 4HF/: x and y and z = (x and y) and z Where E+'D: 1: x**2+17*x*y+4*y**2 where x=1, y=2; 51 2: for i:=1:5 collect x**i*q where q=for j:= product j; {x,2x2,6x3,24x4,120x5} 3: x**2+y+z where z=y**3, y=3; X2+y3+3 EB/'1/GJ J E*:J1 (G H3JDG J '~1'*H1 where *#+J1J (1 EB/'1 'JF E*:J1 .'1, '2 9('1* EH1/F81 F/'1/. list '~1'*H1 list '2 "1HE'FG'J .H/ J DJ3* EJ 3'2/: list (item, item,& ); 1: liss := list (c,b,c, {xx,yy} ,3x**2+7x+3, df (sin(2*x),x)); liss:= {c,b,c,{xx,yy},3x2+7x+3,2cos(2x)} (1'J '1 1/F (' J list '~1'*H1G'J E.*DAJ H,H/ /'1/: (1 /3*H1'* third (list) , second (list) , first (list) (G *1*J( 'DE'F 'HD  /HE H 3HE list 1' (1EJ 1/'FF/. /3*H1 DJ *1 (1'J /3*13J (G 'DE'F G'J part , list '3*: 1: alist := list (c,b,c, {xx, yy} , x**z)$ 2: part (a list, 4) ; {xx, yy} 3: part (a list, 4, 1); xx cons (item, list) (2  EMBED Equation.3  item 1' (G 'HD list '6'AG EJ F/ H E9'/D 9ED1 . (dot) '3*. 1: a list := {b,c}$ 2: a. a list; {a,b,c} 3: liss := cons(a,{b}); % also liss := a cons{b}; new liss := {{a,x}, {b,x}} 5: for each y in new liss sum (first(y))*(second(y)); x(a+b+c) (3 length (list)  EMBED Equation.3  *9/'/ 'DE'F G'J J list 1' EJ /G/. (4 append (list, list)  EMBED Equation.3  /H list 1' JJ EJ F/ ((' GE'F *1*J( "1HE'FG') (5 rest (list)  EMBED Equation.3  DJ3*J 1' (1EJ 1/'F/ G 'DE'F 'HD list '2 "F -0A 4/G '3*. (6 reverse(list)  EMBED Equation.3  (193 list 1' (1EJ 1/'F/. Reverse H cons EJ *H'FF/ (1'J '6'AG 1/F J 'DE'F (G 'F*G'J J DJ3* '3*A'/G 4HF/: 1: reverse (q cons reverse (a,b,c)); {a,b,c,q} (7 sort (list , comp)  EMBED Equation.3  '96'J list 1' (1 -3( B'9/G 'J G (' comp *9JJF EJ 4H/ sort EJ F/: 1: for i:=1:10 collect random(50)$ 2: sort (ws, > =); {41,38,33,30,28,25,20,17,8,5} ws ws ".1JF F*J,G 1' (1EJ 1/'F/ H ws(number) F*J,G J .7J G 4E'1G J "F (1'(1 (' number '3* 1' (1EJ 1/'F/. 1: df (siny,y); cos(y) 2: ws 2; Cos(y)2 3: df (ws 1, y); -sin(y) 3.1.9ED1G'J 1J'6J (Arithmatic operators): ln , log G1 /H '~1'*H1 D'1J*E 7(J9J 1' -3'( EJ FF/. (1'J E-'3(G ('J/ 3HJ  HYPERLINK \l "rounded" rounded 1H4F ('4/. E-/H/J* G'J ln /1 EB'(D log /1 E+'D G'J 2J1 H'6- '3*: 1: ln(4); 1: log(4); ln(4) log(4) 2: ln(e) 2: log(e); ln(e) 1 3: on rounded; 3: on rounded; 4: ln(e); 4: log(e); 1 1 5: ln(4); 5: log(4); 1.38629436112 1.38629436112 6: df(ln(x),x); 6: df(log(x),x);  EMBED Equation.3   EMBED Equation.3  '3*A'/G '2 log *H5JG EJ 4H/. log(expression, integer) D'1J*E expr. 1' /1 E(F'J int. -3'( EJ F/ H E-/H/J* G'J "F E+D ln '3*. Sqrt 1: sqrt (16*  EMBED Equation.3 3);  EMBED Equation.3  2: sqrt (17);  EMBED Equation.3  3: on rounded; 4: sqrt (17); 4: 1231056562 5: sqrt (a*b*c EMBED Equation.3 5*d EMBED Equation.3 3*27); 5: 1961524271 (d3c5ba)0.5 6: off rounded; 7: ws; abs(d)c2 3 EMBED Equation.3  /1 41'J7J G 3HJ  HYPERLINK \l "precise"  precise  1H4F ('4/  EMBED Equation.3 1' (G 5H1* abs(d) FE'J4 EJ /G/ H '1 off ('4/  AB7 .H/ d 1' '~ EJ F/. '~1'*H1G'J 1J'6J /J1J GE G3*F/ G HF 3'/G 'F/ J' '1(1/ 2J'/J F/'1F/ "FG' 1' (G '.*5'1 41- EJ /GJE: (1 abs (expression)  EMBED Equation.3 'F/'2G J expression 1' (1EJ 1/'F/. (2 choose (m,n)  EMBED Equation.3  'F*.'( m '2 n  EMBED Equation.3   EMBED Equation.3  (3 factorial number  EMBED Equation.3  A'*H1JD number 1' -3'( EJ F/. (4 gcd(a,b)  EMBED Equation.3  (.E.E /H 9// a H b 1' -3'( EJ F/. (5 max (expression, expression,& )  EMBED Equation.3  E'3JEE F/ expression 1' EJ /G/. (6 next prime (expression)  EMBED Equation.3  'HDJF 9// 'HD (9/ '2 expression 1' EJ /G/. (7 norm (expression)  EMBED Equation.3  (' 1H4F (H/F 3H&J  HYPERLINK \l "rounded" rounded 'F/'2G J exp. 1' -3'( EJ F/. '1 3H&J  HYPERLINK \l "complex" complex GE 1H4F ('4/  'F/'2G J 9// E.*D7 (a+bi) 1' GE -3'( EJ F/. (8 perm (m,n)  EMBED Equation.3  *1J( m '2 n 1' -3'( EJ F/. (9 remainder (expression1, expression2)  EMBED Equation.3  ('BJE'F/G J *B3JE expr.1 (1 expr.2 1' -3'( EJ F/. expression G' EJ *H'FF/ 9// J' F/ ,EDG 'J ('4F/. (10 round (number)  EMBED Equation.3  number 1' (G F2/J *1JF integer 1/ EJ F/. (11 sign (expression)  EMBED Equation.3  1 J' 0 J' -1 (1EJ 1/'F/ H 9D'E* expr. 1' *9JJF EJ F/. Boolean Operators.3.2: (1 =  EMBED Equation.3  '1 /H 71A (1'(1 ('4F/ true (1EJ 1/'F/. even p (integer) (2  EMBED Equation.3  '1 integer E+(* H 2H, ('4/ true (1EJ 1/'F/. <expression> member <list> (3  EMBED Equation.3  '1 <expr.> 9F51 <list> ('4/ true '3*. <expression> neq <expression> (4  EMBED Equation.3  '1 /H <expr.> (1'(1 F('4F/  true (1EJ 1/'F/. not (logical expression) (5  EMBED Equation.3  '1 (false) nil, logical expr. ('4/ true (1EJ 1/'F/. number p (expression) (6 '1 expr. 9// ('4/ true (1EJ 1/'F/. Primep (expression) (7  EMBED Equation.3  '1 expr. 9// 'HD ('4/  true (1EJ 1/'F/. 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Solve (<equation>,<var>) J': Solve({<equation1>,<equation2>,& },{<var1>,<var2>,& }) DJ3* /HE F4'F E /G/ G E9'/DG (1 -3( /'E E*:J1G' ('J/ -D 4H/ H '1 *9/'/ E9'/D'* H E*:J1 G' (1'(1 ('4/ E *H'F "F 1' -0A 1/. 1: sss := solve(x^2+7); x  EMBED Equation.3  2: rhs first sss;  EMBED Equation.3  3: solve(sin(x^2*y),y);  EMBED Equation.3  4: off allbranch; 5: solve(sin(x^2*y),y); {y=0} 6: solve({3x+5y=-4,2x+y=-10},{x.y});  EMBED Equation.3  7: solve({x+a*y+z,2x+5},{x,y});  EMBED Equation.3  8: ab := (x+2)^2*(x^6+17x+1)$ 9: solve(ab,x);  EMBED Equation.3  10: root_multiplicities; {2,1} '1 9ED1 solve F*H'F/ J  HYPERLINK \l "equation" E9'/DG 1' -D F/, (.4 -D F4/G 1' (G 9FH'F "1HE'F root_of (1E 1/'F/. /1,G  1J4G G' E9'/DG /1 E*:J1 root_multiplicities 0.J1G E 4H/ H (' 5/' 1/F 'JF E*:J1 /1 DJ3* FE'J4 /'/G E 4H/. '1 J /3*'G E9'/D'* .7 /'4*G ('4JE (' 1H4F 1/F 3H&J  HYPERLINK \l "cramer" cramer 319* -D E9'/D'* 'A2'J4 E J'(/. '1 E9'/D'* J /3*'G E*F'B6 ('4F/ solve J DJ3* .'D (1E 1/'F/ H '1 'JF E9'/D'* ~'1'E*1 ('4F/, E*:J1 requirements 41'J7 1' G *-* "F E9'/D'* 3'2'1 .H'GF/4/ 1' F4'F E /G/. 417 3'2'1 ('J/ E3'H 5A1 B1'1 (J1/: 1: solve({x-a,x-y,y-1},{x,y}); {} 2: requirements; {a-1} (1' J /3*'G E9'/D'* .7 H ~'1'E*1 E*:J1 assumptions DJ3* '2 1H'(7 (1' 'JF ~'1'E*1G' 1' (1E 1/'F/ H ,H'( E9'/DG *' HB* E9*(1 '3* G GJ /'E '2 'JF 1H'(7 E3'H 5A1 F4HF/: 1: solve({a*x-b*y+x,y-c},{x,y});  EMBED Equation.3  2: assumptions; {a+1} HB* 3H&J allbranch 1H4F '3* ( defult ) *E'E 1J4G G' E9'/DG F4'F /'/G E 4HF/. HD '1 .'EH4 ('4/, AB7 1J4G G' '5D F4'F /'/G E 4HF/. E+D .7 4H5 E+'D 'HD. Ode solve ode solve package (1'J -D (ordinary differential equation) ode '3* H E-/H/J* G'JJ /'1/: -1 AB7 J E9'/DG 1' EJ *H'F/ -D F/. -2 AB7 B'/1 (G -D E9'/DG 3'/G J E1*(G 'HD  E9'/DG J .7J (' 61'J( +'(* H E9'/D'* 'HJD1 '3*. ode solve (expr., var1, var2) expr. =0 E9'/DG /JA1'F3JD EH1/ F81 '3*. var1 E*:J1 H'(3*G Hvar2 E*:J1 E3*BD '3*. /1 expr.  E4*B 1' (' 9ED1  HYPERLINK \l "df" df FE'J4 EJ /GJE. 1: depend y,x ; 2: ode: = df (y,x) +y*sinx/cosx-1/cosx $ 3: ode solve (ode, y, x); {y=arbconst(1)*cos(x)+sin(x)} '~1'*H1 arbconst E.AA arbitrary constant '3* (1'J FE'J4 +'(* /D.H'G /1 -D E9'/DG /JA1'F3JD Show rules *E'E BH'FJF E1(H7 (G "1HE'F .H/ 1' FE'J4 EJ /G/: 1: show rules log;  EMBED Equation.3  'JF BH'9/ E+D '96'J J  HYPERLINK \l "list" list B'(D /3*13 G3*F/: 2: rhs second ws; 1 (1.J '~1'*H1G'J ,(1J /J1: Conj (expression) (1  EMBED Equation.3  E2/H, E.*D7 expression 1' (1EJ 1/'F/. Hypot (n,m) (2  EMBED Equation.3  '1 m, n 9// ('4F/  EMBED Equation.3 1' (1EJ 1/'F/. Import (expression) (3  EMBED Equation.3  (.4 EHGHEJ expression 1' (1EJ 1/'F/. Repart (expression) (4  EMBED Equation.3  (.4 -BJBJ expression 1' (1EJ 1/'F/. num (expression) (5  EMBED Equation.3  (9/ '2 3'/G 1/F expression 5H1* "F 1' (1EJ 1/'F/. prod (expression, k, lolim, uplim) (6  EMBED Equation.3  E9'/D  EMBED Equation.3 '3*. sum(expression, k, lolim, uplim) (7  EMBED Equation.3  E9'/D  EMBED Equation.3 '3*. 4.*H'(9 1J'6J exp exp (expression) or  EMBED Equation.3  or e** 1: exp (sin(x)) ; esin(x) 2: exp (11) ; e11 3: on nonded 4: exp (sin(pi/3)); 2.37744267524 (1'J 1A*F ,H'( 51J- 9//J  3HJ  HYPERLINK \l "rounded" rounded 1H4F ('4/. erf *'(9 .7': erf(x)= (2/sqrt(pi))*int( EMBED Equation.3 (- EMBED Equation.3 2),x) .H'5 E-/H/J '2 'JF *'(9 (1'J REDUCE 4F'.*G 4/G '3* '2 ,EDG A1/ (H/F H E4*B "F H '2 1HJ 'JFG' (96J 'F*1'DG' B'(D E-'3(G '3*. 1:erf(0) ; 0 2: erf(-a) ; -erf(a) 3: df (erf( EMBED Equation.3 2),x) ;  EMBED Equation.3  4: int (erf(x),x) ;  EMBED Equation.3  *H'(9 E+D+'*J *H'(9 E+D+'*J (' '~1'*H1 GE F'E .H/ B'(D /3*13J G3*F/: cos(expr), cosh(expr), cot(expr), coth(expr), csc(expr), csch(expr), ses(expr), sech(expr), sin(expr), sinh(expr), tan(expr), tanh(expr) E9H3 'JF *H'(9 (& , sin-1, cos-1) (' '6'AG 1/F "a" (G 'HD F'E '~1'*H1G' (G '1 EJ 1HF/: acos(expr), acosh(expr), acot(expr), acoth(expr), acsc(expr), acsch(expr), asec(expr), asech(expr), asin(expr), asinh(expr), atan(expr), atanh(expr) (1'J 1A*F ,H'( /BJB H 51J- 9//J  3H&J,  HYPERLINK \l "rounded" rounded ('J/ 1H4F ('4/. *H'(9 .'5 1J'6J /1 Special Function Package (' /3*H1  HYPERLINK \l "load" load-package specfn B'(D /3*13J G3*F/. '7D'9'* EH,H/ /1 'JF package: (1.J +'(* G'J E91HA  *'(9 (3D  F/ ,EDG 'J G'J .'5  *'(9 'E'  *'(9 2*' (Zeta)  *H'(9 H 'F*1'D G'J (J6HJ  61'J( Clebsch-Gordan H ... (1'J G1 /'E '2 'JFG' F/ '~1'*H1 H,H/ /'1/ G E91AJ GEG J "FG' /1 'JF 1'GFE'J .D'5G FEJ F,/. (' '3*A'/G '2 'JF package (96J E-'3('* GE 1'-**1 EJ 4H/. E+D E-'3(G J 31JG'JJ G ,H'( "FG' 4'ED *H'(9 .'5J '3*. 5.Declaration array *91JA "1'JG (G 'JF 4D '3*: array identifier (dimension, dimension, & ) G1 ,2! "1'JG (D'A'5DG (9/ '2 *91JA "F  (' 5A1 EB/'1/GJ EJ 4H/. 1: array a(2,5), b(3,3,3); 2: a(3,4); 0 3: length a; {3,6} *91JA "1'JG GEJ4G global '3*. -*J '1 /'.D  HYPERLINK \l "procedure" procedure J'  HYPERLINK \l "block" block *91JA 4H/. ',2'J "1'JG (G 5H1* a(i,j) B'(D /3*13J G3*F/. '~1'*H1  HYPERLINK \l "assign" : = J' /3*H1  HYPERLINK \l "let" let AB7 (1'J * * ',2'J "1'JG '1(1/ /'1/  FG D "1'JG. 'F/J3 "1'JG J' 4E'1G J .'FG G'J "F '2 5A1 41H9 EJ 4H/. (F'(1'JF E+D' .'FG G'J a(5) '2 5A1 *' 5 4E'1G EJ .H1F/. ~3 7HD "1'JG J 6, (length a)a '3*. Precision /B* '94'1 1' /1 -'D*J G 3H&J  HYPERLINK \l "rounded" rounded 1H4F '3* *9JJF EJ F/ (~J4 A16 12 '3*) 1: on rounded; 2: 7/9; 0.777777777778 3: precision 20; 20 4: 7/9; 0.77777777777777777778 depend HB*J /1 *91JA '2 depend '3*A'/G EJ FJE  J9FJ argument 'HD (3*J (G (BJG J argument G' /'1/. 'JF (3*J (' '3*A'/G '2 nodepend '2 (JF EJ 1H/. 1: depend y,x ; 1: depend z, cos(x), y ; 2: df (y**2, x) ; 2: df (sin(z) , cos(x) ;  EMBED Equation.3   EMBED Equation.3  3: nodepend y,x ; 3: depend y, x ; 4: df (y**z, x) ; 4: df ( EMBED Equation.3 z , x); 0  EMBED Equation.3  '~1'*H1G'J .7J  /1 '9E'D .7J (H/F H'(3*J E*:J1G' 1' /1 F81 EJ J1F/ ((G *91JA .7J '~1'*H1G' E1',9G FJ/) (' '3*A'/G '2 '~1'*H1  HYPERLINK \l "freeof" free of EJ *H'FJ/ H'(3*J /H E*:J1 1'  FJ/. (J4 *1JF '1(1/ depend /1 -D E9'/DG /JA1'F3JD (' '3*A'/G '2 9ED1  HYPERLINK \l "odesolve" ode solve '3*. Operator (' *91JA 1/F J '~1'*H1 EJ *H'FJ/ 9ED1 /D.H'G .H/ 1' (3'2J/. 1: operator dis ; 2: let dis (~x, ~y) = sqrt ( EMBED Equation.3 ; 3: dis (1,2) ;  EMBED Equation.3  4: dis (a,10);  EMBED Equation.3  1: operator fac; 2: let fac(~n) = if n=0 then 1 2: else if not (fixp n and n>0) 2: then rederr uchoose non-negative integer" 3: fac (5); 120 EH'8( ('4J/ JJ '2 '~1'*H1G'J .H/ 3J3*E 1' /H('1G *91JA FFJ/. '~1'*H1G' EJ *H'FF/ /'.D  HYPERLINK \l "procedure" procedure *91JA 4HF/ H /1 'JF 5H1* global G3*F/. ~'1'E*1 5H1J (5/' 2FF/G J procedure) EJ *H'F/ J '~1'*H1 ('4/. '~1'*H1G' EJ *H'FF/ (' .H'5 E.*DAJ *91JA 4HF/: (1 infix  EMBED Equation.3 (1'J *91JA 9ED1 EJ'FHF/J. 1: infix aa; 2: for all x, y let aa(x,y). cos(x)*cos(y)-sin(x)*sin(y); 3: x aa y; cos(x)cos(y)-sin(x)sin(y) 4: pi/3 aa pi/2 ;  EMBED Equation.3  /B* FJ/ G /1 9('1*  HYPERLINK \l "let" let  '~1'*H1 ~J4HF/J *91JA 4/G '3*. (9/ '2 *91JA '~1'*H1 EJ *H'F/ GE (G 5H1* ~J4HF/J H GE EJ'FHF/J '3*A'/G 4H/. (2 even: '~1'*H1 F3(* (G "1HE'F 'HD4 A1/ *91JA EJ F/: 1: even f; 2: f(-a, -b); f(a, -b) linear (3  EMBED Equation.3 J '~1'*H1 1' (1 -3( E*:J1 'HD "F .7J *91JA EJ F/. 'HD '~1'*H1 ('J/ *91JA 4/G ('4/ H (9/ .7J *91JA 4H/. 1: operator f; 2: linear f ; 3: f(o,x) ; 0 4: f(-y,x) ; -f(l,x)y 5: f(y+z,x) ; f(1,x)(y+z) 6: depend z,x ; 7: f(y*z, x) ; f(z, x)y 8: depend y,x ; 9: f( EMBED Equation.3 ) ; f( EMBED Equation.3 ) 10: nondepend z,x ; 11: f( EMBED Equation.3 ) ;  EMBED Equation.3  /B* FJ/ G '6'AG 1/F H'(3*J E*:J1G' HFG /1 '9E'D .7J (H/F *#+J1 EJ 0'1F/. '1 argument 'HD 4'ED argument /HE ('4/ GE *#+J1 'JF H'(3*J (G GEJF 4D '9E'D EJ 4H/. (4 noncom  EMBED Equation.3  '1 J '~1'*H1 1' nancom *91JA FJE (G 'JF E9FJ '3* G 'JF '~1'*H1 *-* 9ED 61( ,' (G ,' ~0J1 FJ3*: 1: operator f, h ; 2: noncom f ; 3: h(a)* h(b)-h(b)* h(a) ; 0 4: f(u)* f(b)-f(b)* f(a); f(a)f(b)-f(b)f(a) /1 ~J4 A16 REDUCE J '~1'*H1  :J1.7J  ~J4HF/J H ,'(G ,' ~0J1 *91JA EJ 4H/. scalar, real, integer *91JA E*:J1 (G 5H1* integer, real J' scalar ('J/ (D'A'5DG (9/ '2 begin /1 J  HYPERLINK \l "block" block 'F,'E 4H/. *91JA  (G 5H1* local /'.D block '3* H (D'A'5DG (' 5A1 EB/'1/GJ EJ 4H/. (9/ '2 ~'J'F block 'JF E*:J1 -0A EJ 4H/. *:JJ1'* EB/'1 "F /'.D block E*:J1 GE F'E "F .'1, '2 block 1' *:JJ1 FEJ /G/. '3*A'/G '2 real H integer (G E9FJ 'JFG REDUCE FH9 E*:J1 1'  EJ F/ FJ3* H AB7 (1'J '7D'9 .H/ '1(1 '3*. equation equation 9('1* '3* G /1 "F /H 9('1* ,(1 (G H3JDG  9ED1 = (G GE H5D 4/G 'F/ H (G 4D 2J1 '3*: left-hand side=right-hand side 9('1* 3E* ~ E9'/DG (G H3JDG  9ED1 lhs(<equation>) H 9('1* 3E* 1'3* "F (G H3JDG  9ED1 rhs(<equation>) B'(D /3*13 G3*F/. 9('1* 3E* 1'3* GEJ4G E-'3(G E 4H/. HD (1' E-'3(G 4/F 9('1* 3E* ~, 3H&J evallhseqp ('J/ 1H4F ('4/. HB* J E9'/DG (.4 '2 J 9('1* EF7B '3*, E+D 9('1*  HYPERLINK \l "if" if J'  HYPERLINK \l "while" while , /H 71A "F '2 GE E 4/G H EB/'1 "F (' 5A1 EB'J3G E 4H/. E9'/DG E *H'F/ J 96H  HYPERLINK \l "list" list ('4/. GEFJF E *H'F J E*:J1 1' (1'(1 (' J equation B1'1 /'/. '1 3H&J evallhseqp 1H4F ('4/, E *H'F /H E9'/DG 1' (' GE ,E9 J' '2 GE E 1/ J' J E9'/DG 1' (G *H'F 13'F/. 1: on evallhseqp; 2: u := x+y=1$ 3: v := 2x-y=0$ 4: 2*u-v; -3y=-2 5: ws/3;  EMBED Equation.3  6: lhs(u); x+y 6.3H&J G' (switches ) 3HJ G' EJ *H'FF/ (G H3JDG J /3*H1G'J on H off 1H4F H .'EH4 4HF/: on switch-name ; off switch-name ; 1H4F J' .'EH4 (H/F J 3H&J '1'JJ 3J3*E 1' *:JJ1 EJ /G/. *9/'/ 3H&J G'J *91JA 4/G /1 REDUCE (3J'1 2J'/ '3*. /1 'JF,' AB7 (G 3H&J G'JJ G /1 E+'D G' (G "FG' '4'1G 1/JE J' EGE *1 G3*F/ EJ ~1/'2JE: algint '1(1/ 3H&J algint 1' /1 '3*A'/G '2 '~1'*H1  HYPERLINK \l "int" int ((JFJ/. combineexpt /1 5H1* 1H4F (H/F  *H'F'JJ REDUCE 1' /1 3'/G 1/F 9('1*G'J *H'FJ 'A2'J4 EJ /G/: 1:  EMBED Equation.3   EMBED Equation.3  2: on combineexpt ; 3: ws ; 3 complex (' 1H4F (H/F complex BH'FJF 1J'6J'* E.*D7 /1 3'/G 3'2J  E-'3(G J *H'(9  A'*H1J1J H ... '3*A'/G EJ 4H/ (/HF "F  REDUCE *FG' EJ /'F/ G i (1'(1 ('  EMBED Equation.3  '3*: 1: ( EMBED Equation.3 + EMBED Equation.3 2)/(x+i*y) ; (iy+x)-1(x2+y2) 2: on complex ; 3: ws ; x-iy cramer (' 1H4F (H/F 3H&J cramer  9EDJ'* E9H3 1/F HYPERLINK \l "matrix"  E'*1J3 H -D E9'/D'* .7J ( HYPERLINK \l "solve" solve) (' '3*A'/G '2 1H4 1'E12 'F,'E EJ 4H/ G 31J9 *1 '3*. 'D(*G '1 /1'JG G'J E'*1J3 9// ('4F/ 9EDJ'* F/*1 EJ4H/ (F'(1'JF B(D '2 1H4F 1/F 'JF 3H&J (G FH9 E'*1J3 *H,G FJ/. failhard /1 5H1* 1H4F (H/F 'JF 3H&J  /1 5H1*J G REDUCE F*H'F/ J 'F*1'D 1' (G A1E (3*G -3'( F/  ~J:'E .7' EJ /G/. gcd '1 'JF 3H&J 1H4F ('4/  A'*H1G'J E4*1 5H1* H E.1, GF'E '~ ,H'( 3'/G EJ 4HF/. listargs /1 -'D* DJ  '1 "1HE'F J '~1'*H1  HYPERLINK \l "list" list ('4/ HDJ 'JF '~1'*H1 (1'J  HYPERLINK \l "list" list *91JA F4/G ('4/  ,H'(  DJ3*J '3* G /1 96H "F *#+J1 '~1'*H1 1HJ 96H E*F'81 "F /1 "1HE'F *'(9 '3*. 1H4F (H/ 3H&J listargs '2 'JF 9ED ,DHJ1J EJ F/: 1: log {a,b,c} ; {log(a), log(b), log(c)} 2: on listargs ; 3: log {a,b,c} ; log({a,b,c}) multiplicities /1 -D J  HYPERLINK \l "equation" E9'/DG ( HYPERLINK \l "solve" solve) '1 J 1J4G /1,G 'J (J4*1 '2 J /'4*G ('4/  51J-'K /1 ,H'( 'JF /1,G F4'F /'/G FEJ 4H/. (DG /1 E*:J1  HYPERLINK \l "root" root-multiplicities 0.J1G EJ 4H/  HDJ (' 1H4F (H/F 3H&J multiplicities  /1,G J 1J4G /1 .H/ ,H'( E4.5 '3*: 1: solve( EMBED Equation.3 2=2x-1, x) ; x=1 2: root-multiplicities ; 2 3: on multiplicities; 4: solve ( EMBED Equation.3 2=2x-1, x); x=1 , x=1 nero '1 3H&J nero 1H4F ('4/  E*:J1J G EB/'1 5A1 /'1/ '~ FEJ 4H/. 'JF '1 (G .5H5 HB*J (' E'*1J3 G'J (21 31 H '1 /'1JE EAJ/ '3* G EJ *H'F AB7 /1'JG G'J :J1 5A1 1' /1 5A-G /J/. output '1 3H&J output .'EH4 ('4/  GJ F*J,G 'J 1HJ 5A-G '~ FEJ 4H/ -*J '1 'F*G'J .7 /3*H1 ; ('4/. HF '~ F*J,G 2E'F (1 '3*  /1 5H1*J G (' E-'3('* EA5D H F*'J, ~JJ/G 31H'1 /'1J/  .'EH4 1/F 'JF 3H&J EJ *H'F/ (1F'EG 1' 31J9 *1 ~J4 ((1/. /1 5H1* '-*J', (G F*J,G J J .7 .'5 EJ *H'FJ/ '2  HYPERLINK \l "ws" ws(number) '3*A'/G FJ/. Precise 1: sqrt ( EMBED Equation.3 2); x 2:(  EMBED Equation.3 2 EMBED Equation.3 (1/4);  EMBED Equation.3  3: on precise; 4: sqrt (  EMBED Equation.3 2 EMBED Equation.3 ; abs(s) 5: (  EMBED Equation.3 2 EMBED Equation.3 (1/4);  EMBED Equation.3  Rounded, roundall, roundbf (' 1H4F (H/F 3H&J rounded 1J'6J'* '94'1J A9'D EJ 4H/. 1: pi ;  EMBED Equation.3  2: 35/217 ;  EMBED Equation.3  3: on rounded ; 4: pi ; 3.141592359 5: 35/217 ; 0.161 (1'J *:JJ1 /B* '94'1 '2 /3*H1 precision '3*A'/G FJ/. (G 7H1 ~J4 A16  /B* '94'1 12 1BE '3*. '1 (.H'GJ/ '9/'/ 31J (G 4D '94'1J /1 FJ'JF/  3H&J round all 1' .'EH4 FJ/: 1: on rounded ; 2: sqrt (3); 1.73205080756 3: 1/2 ; 0.5 4: off round all ; 5: 1/2 ;  EMBED Equation.3  '1 (.H'GJ/ REDUCE '9/'/ (3J'1 H 1' 5A1 '~ FF/ ('J/ 3H&J roundbf 1' 1H4F FJ/. 1: on rounded; 2: exp (-100000. EMBED Equation.3 2); 0 3: on round bf ; 4: exp (-100000.  EMBED Equation.3 2); 1.18441281937e-4342953505 Time '1 3H&J time 1H4F ('4/  2E'F 3J3*E G (1'J ',1' 1/F G1 2'1G J REDUCE '3*A'/G 4/G '3* (9/ '2 '~ ,H'(  FE'J4 /'/G EJ 4H/. 1: on time ; time : 281 ms 2: df (sin ( EMBED Equation.3 2+y), y); cos( EMBED Equation.3 +y) time: 0 ms 3: solve ( EMBED Equation.3 2-6*y, x);  EMBED Equation.3  Time: 16 ms 2E'FJ G (9/ '2 1H4F 1/F time '~ EJ 4H/  2E'FJ '3* G '2 '(*/'J 41H9 (1F'EG E-'3(G 4/G '3*. (9/ '2 G1 /3*H1  2E'F E-'3(G '~ EJ 4H/ G 4'ED 2E'F *'J~ 1/F /3*H1 FJ3*. /1 ,'G'JJ G '/9' 1/JE 1H4F (H/F J 3H&J EJ *H'F/ 319* ',1' H E-'3(G 1' 'A2'J4 /G/ (E+D 3H&J  HYPERLINK \l "cramer" cramer H  HYPERLINK \l "output" output) 'JF 319* 1' EJ *H'FJE (' 1H4F 1/F time  FJE. 7.E'*1J3 G'( Matrix ) *91JA E'*1J3: J  HYPERLINK \l "identifier" identifier 1' EJ *H'FJE '2 FH9 matrix *91JA FJE. 1: matrix a, b(1,4); 2: b(1,1) ; 0 3: a(1,1) ; ***** Matrix A not set *9JJF E1*(G J E'*1J3 GF'E *91JA "F ',('1J FJ3*. HDJ /1 5H1* *91JA E1*(G GEG J /1'JG G' (' 5A1 EB/'1/GJ EJ 4HF/ H /1 5H1* 9/E *91JA E1*(G  /1'JG G' B'(D /3*13J FJ3*F/. 4: a:= mat[(x0, y0), (x1, y1)) 5: length a; {2,2} 6: b: =  EMBED Equation.3 2 ;  EMBED Equation.3  /1 5H1* FJ'2  E1*(G J E'*1J3 (' E-'3('* GE'GF EJ 4H/ (E+D E'*1J3 b)  HYPERLINK \l "let" let 1' EJ *H'F (G D J E'*1J3 '9E'D 1/  HDJ '1  HYPERLINK \l "let" let H  HYPERLINK \l "clear" clear (G J /1'JG J E'*1J3 (.H'G/ '9E'D 4H/  -*E'K 'JF /1'JG ('J/ E-*HJ J +'(* ('4/. 9EDJ'* ,(1J 1' EJ *H'F (1'J E'*1J3 G' (G '1 (1/ /1 5H1*J G E1*(G G' /13* ('4/ (+ - * /) 1/A J' /A 93 E'*1J3 A 1' EJ /G/ (/1 5H1* E9H3 ~0J1J). A/B GE (G 5H1*  EMBED Equation.3  *91JA EJ 4H/. GE FJF E'*1J3 G'J E1(9J 1' EJ *H'F (G *H'F 5-J- E+(* H EFAJ (/1 5H1* E9H3 ~0J1J) 13'F/. '~1'*H1 mat : (1'J FE'J4 E'*1J3 1: mat ((1,2), (3,4)) ;  EMBED Equation.3  2: matrix q; 3: a;= ws ; q :=  EMBED Equation.3  4: a: = mat ((sin(x), cos(x), 1); a:= (sin(x) cos(x) 1) (1'J '3*A'/G '2 mat '-*J',J FJ3* G E'*1J3 B(D'K *91JA 4/G ('4/ H E1*(G "F E4.5 ('4/. Mat /1 5H1* FJ'2 E1*(G J *91JA 4/G (1'J E'*1J3 1' *:JJ1 EJ /G/. '1 (.H'GJE (G /1'JG G' /3*13J /'4*G ('4JE (a(i,j)) -*E'K E'*1J3 ('J/ *91JA 4/G ('4/. '~1'*H1 cofactor cofactor (matrix-expression, row, column) 1: cofactor (mat ((a,b,c), (d,e,f), (p,q,r)), 2, 2); ar- cp 2: cofactor (mat((a,b,c), (d,e,f)), 2, 2); ***** non-square matrix '~1'*H1 def : (1'J E-'3(G J /*1EJF'F E'*1J3 1: m: = mat (a,b,c), (d,e,f), (p,q,r)) $ 2: def m; ,% also det(m); aer-afq-bdr+bfp+cdq-cep 3: def (5); 5 '~1'*H1 mateigen: (1'J (G /3* "H1/F E9'/DG J HJG EB/'1 H HJG (1/'1G'J J E'*1J3 mateigen (matrix-expression, var) Var F'E E*:JJ1 E9'/DG J HJG EB/'1 '3*. 1: aa: = mat (2,5), (1,0)) $ 2: mateigen (aa, alpha);  EMBED Equation.3  3: bb: = mat ((1,2,3), (4,5,6), (7,8,9))$ 4: mateigen (bb, alpha);  EMBED Equation.3  /1 DJ3* G'JJ G (11/'F/G EJ 4H/  96H 'HD E9'/DG J HJG EB/'1 '3* G E3'HJ 5A1B1'1 EJ J1/  96H /HE /1,G J HJG EB/'1 H 96H 3HE HJG (1/'1 E*F'81 '3*. '~1'*H1 arbcomplex /1 J expression (J'F1 J (.4 '3'D1 /D.H'G /1 "F expression '3*. '~1'*H1 nullspace: (1'J E'*1J3 a DJ3*J '2 (1/'1G'J E3*BD .7J 1' -3'( EJ F/ G *1J( .7J "FG' E9'/DG J ax=0 1' '16' EJ F/. nullspace(matrix-expression) 1: nullspace(mat((1,2,3,4), (5,6,7,8)));  EMBED Equation.3  '~1'*H1 smithex: A1E F1E'D smith 1' (1'J E'*1J3 A -3'( EJ F/ H  EMBED Equation.3  1' (1EJ 1/'F/ G :  EMBED Equation.3  smithex (<matrix>,<variable>) <matrix> E'*1J3J (1 -3( <variable> '3*. 1: a: = mat ((x,x+1), (0,3* EMBED Equation.3 2))$ 2: smithex (a,x);  EMBED Equation.3  '~1'*H1G'J /J1J '2 ,EDG smithex-int(<matrix>)  frobenius(<matrix>)  jordan(<matrix>) H ... FJ2 '11/ E4'(GJ /'1F/ G (G *1*J( A1E F1E'D smith (1'J E'*1J3 (' /1'JG G'J integer  A1E F1E'D frobenius H A1E F1E'D Jordan 1' (1'J E'*1J3 A -3'( EJ FF/ H (G *1*J(  EMBED Equation.3    EMBED Equation.3 H  EMBED Equation.3 1' (1EJ 1/'FF/ G  EMBED Equation.3   EMBED Equation.3 . add-columns (<matrix>, <c1>, <c2>, <expr>) : (G 3*HF c2 '2 <matrix> c1*<exper> 1' '6'AG EJ F/. add-rows GE '11/ E4'(GJ (1'J 371G' /'1/. add-to-columns (<matrix>, <column-list>, <expr>) : (G 3*HF G'JJ G /1 DJ3* <column-list> E4.5 4/G 'F/  <expr> 1' '6'AG EJ F/. add-to-rows GE '11/ E4'(GJ (1'J 371G /'1/. /H '~1'*H1 mult-columns H mult-rows (' syntax E4'(G GEJF '1 1' (' 9EDJ'* 61( ((G ,'J ,E9) 'F,'E EJ /GF/. augment-comlumns (<matrix>, <column-list>): 3*HF G'J E4.5 4/G /1 D3J* <column-list> '2 <matrix> 1' FG EJ /'1/ H /1 E'*1J3 ,/J/J F'1 GE B1'1 EJ /G/. Stack-rows GEJF '1 1' (1'J 371G' 'F,'E EJ /G/. band-matrix (<expr-list>,<size>) : <expr-list> 3JFJ (' *9/'/ '96'J A1/ '3* H band matrix J E'*1J3 E1(9J (' 3'J2 <size> EJ 3'2/ H 'JF DJ3* 1' 7H1J /1 371G'J "F B1'1 EJ /G/ G 96H H37 GEJ4G 1HJ B71 ('4/: 1: band-matrixs ({x,y,z},4);  EMBED Equation.3  Char-matrix(<matrix>,<lambda>) : E'*1J3 E4.5G J <matrix> 1' (1 -3( E*:J1 <lambda> EJ 3'2/ J9FJ c=<lambda>*Id-<matrix> G Id E'*1J3 H'-/ '3*. Char-poly (<matrix>, <lambda>): E9'/DG J E4.5G J <matrix> 1' (1 -3( E*:J1 <lambda> ~J/' EJ F/ J9FJ /*1EJF'F c=<lambd>*Id-<matrix> G Id E'*1J3 H'-/ '3*. Cholesky (<matrix>) : *,2JG J cholesky 1' (1'J <matrix> FE'J4 EJ /G/. J2J G (1EJ 1/'F/ {L,U} '3* (G 7H1J G A=LU H  EMBED Equation.3 . /1 'JF '~1'*H1 /1'JG G'J E'*1J3 ('J/ -BJBJ ('4F/. (1'J E'*1J3 (' /1'JG G'J E.*D7 /3*H1 lu-decom(<matrix>) 1' (G '1 EJ (1JE. 1: f: = mat((1,1,0), (1,3,1), (0,1,1)) $ 2: on rounded; 3: cholesky (f);  EMBED Equation.3  Coeff-matrix ({<lineq-list>}) : <lineq-list> DJ3*J '2 E9'/D'* '3*. 'JF '~1'*H1 {C,X,B} 1' (1EJ 1/'F/ (G 7H1J G CX=B 1: coeff-matrix ({x+y+4*z=10, y+x-z=20, x+y+4});  EMBED Equation.3  Row-dim(<matrix>), column-dim (<matrix>)  '(9'/ 3*HF H 371 <matrix> 1' -3'( EJ F/. diagonal (<mat-list>) : E'*1J3J EJ 3'2/ G '96'J <mat-list> 1HJ B71 "F B1'1 /'1F/. '96'J <mat-list> EEF '3* '3'D1 J' E'*1J3 ('4F/. extend (<matrix>, <r>, <c>, <expr>) : E'*1J3 1' (G 'F/'2G J <r> 371 H <c> 3*HF 3*14 EJ /G/ H /1'JG G'J ,/J/ 1' (1'(1 (' <expr> B1'1 EJ /G/. get-columns (<matrix>, <column-list>): 3*HF G'JJ G 4E'1G J "FG' /1 DJ3* <column-list> E4.5 4/G 1' (G 5H1* DJ3*J '2 E'*1J3 G'J 3*HFJ (1EJ 1/'F/. Ger-rows(matrix>,<row-list>) GEJF '1 1' (1'J 371G' EJ F/. tp(<matrix>): *1'FG'/G J <matrix> 1' -3'( EJ F/. hermitian-tp(<matrix>): (1'J E'*1J3 A  A+ 1' -3'( EJ F/. hesian(<expr>, <var-list>): E'*1J3 hessian 1' (1'J <expr> E-'3(G EJ F/ H "F E'*1J3 nn '3* (n: *9/'/ '96'J <var-list>) G /1'JG J (i,j) "F (1'(1 '3* (' df(<expr>,<var-list>(i), <var-list> (i) . hilbert(<size>,<expr>): E'*1J3 E1(9J Hilbert 1' (1'J <expr> E-'3(G EJ F/ H "F E'*1J3 E*B'1FJ '3* G /1'JG J (i,j) "F (1'(1 '3* ('  EMBED Equation.3  jacobian (<expr-list>,<var-list>): E'*1J3 'H(J 1' (1'J <expr-list> ((1 -3( E*:J1G'J (<var-list> E-'3(G EJ F/ G J E'*1J3 mn '3*: n *9/'/ E*:J1G' H m *9/'/ expression G'. /1'JG J (i,j) "F (1'(1 '3* (': df(<expr-list>(i),<var-list>(j)) 1: jacobian EMBED Equation.3   EMBED Equation.3  Jordan-block (<expr>,(size>) : E'*1J3 E1(9J Jordan block 1' (' '(9'/ <size> EJ 3'2/ /1'JG G'J 'JF E'*1J3 (G 'JF 4D G3*F/: J(i,j) = <expr> for i=1,& ,n , J(I,i+1)=1 for i=1,& ,n-1\ H (BJG J /1'JG G' 5A1 G3*F/. make-identity (<size>): E'*1J3 H'-/ 1' (' '(9'/ <size> <size> EJ 3'2/. matrix-augment(<matrix-list>) : E'*1J3 G'J <matrix-list> 1' 'ABJ F'1 GE B1'1 EJ /G/ H J E'*1J3 EJ 3'2/. matrix-stack GEJF '1 1' (G 5H1* 9EH/J 'F,'E EJ /G/. Matrix(<input>): '1 <input> E'*1J3 ('4/ true (1 EJ 1/'F/ H /1 :J1 'JF 5H1* nil. Minor (<matrix>, <r>, <c>) : E'*1J3 ,/J/J EJ 3'2/ 371 <r> H 3*HF <c> '2 <matrix> 1' /1 "F -0A 1/G '3*. random-matrix(<r>,<c>,<limit>) : J E'*1J3 <c><r> EJ 3'2/ G /1'JG G'J "F '9/'/ 1F/HE (JF -<limit> H <limit> G3*F/.  HYPERLINK \l "switch" 3HJ G'J 2J1 FH9 /1'JG G' 1' *9JJF EJ FF/: imaginary (1 : '1 1H4F ('4/ /1'JG G' (G 5H1* x+iy .H'GF/ (H/ G  limit<x,y<limit not-negative (2: '1 1H4F ('4/ /1'JG G' (J' x,y /1 -'D* imaginary) (JF 0 H limit .H'GF/ (H/. only-integer (3: '1 1H4F ('4/ /1'JG G' (J' x,y /1 -'D* imaginary) integer .H'GF/ (H/. symmetric (4 : '1 1H4F ('4/  E'*1J3 E*B'1F .H'G/ (H/. upper-matrix (5: '1 1H4F ('4/  E'*1J3 ('D' E+D+J .H'G/ (H/. lower-matrix (6 : '1 1H4F ('4/  E'*1J3 ~'JJF E+D+J .H'G/ (H/. remove-columns (<matrix>,<column-list>) : E'*1J3 ,/J/J EJ 3'2/ G 3*HF G'J E4.5 4/G /1 <column-list> '2 <matrix> 1' /1 "F -0A 1/G '3*. remove-rows 9EDJ'* E4'(G 1' (1'J 371G' 'F,'E EJ /G/. Squarep (<matrix>) : '1 <matrix> E1(9J ('4/ true (1EJ 1/'F/ H /1 :J1 'JF 5H1* nil sub-matrix(<matrix>, <row-list>, <column-list>) : 3*HF G'J<c2>, <c1> '2 <matrix> 1' (' GE ,' (G ,' EJ F/. swap-rows GEJF '1 1' (1'J 371G' 'F,'E EJ /G/. Swap-entries (<matrix>, <r1,c1> , <r2,c2>) : /1'JG G'J (<r2> , <c2>) , (<r1>, <c1>) 1' (' GE ,' (G ,' EJ F/. Symmetricp (<matrix>) : '1 <matrix) E*B'1F ('4/ true (1EJ 1/'F/ H /1 :J1 'JF 5H1* nil toeplitz (<expr-list>) : '2 DJ3* <expr-list> E'*1J3 toeplitz 1' EJ 3'2/. (G 'JF 4D G 'HDJF expression 1HJ B71 B1'1 EJ J1/ H i- 'EJF expression 1HJ (i-1) 1/JA ('D'*1 H ~'JJF *1 '2 B71: 1: toeplitz ({w,x,y,z});  EMBED Equation.3  Vandermonde({<expr-list>}) : '2 1HJ <expr-list> E'*1J3 vandermonde 1' EJ 3'2/ (G 'JF 4D G /1'JG J (i,j) 1' (1'(1  EMBED Equation.3  B1'1 EJ /G/. '(9'/ E'*1J3 nn '3* G n *9/'/ expression '3*: 1: vander monde ({x,z*y, 3*z});  EMBED Equation.3  Trace (<square-matrix>) : 1/ J E'*1J3 E1(9J 1' E-'3(G EJ F/. Pseudo-inverse (<matrix>) : pseudo inverse (4(G H'1HF) <matrix> 1' E-'3(G EJ F/ G: <matrix>*pseudo-inverse (<matrix>)=Id H D2HEJ F/'1/ G <matrix> J E'*1J3 E1(9J ('4/. J F*G: (G F81 E 13/ /1 J  HYPERLINK \l "group" group J'  HYPERLINK \l "block" block J' J 9('1* *FG' ( E+D'K (/FG   HYPERLINK \l "procedure" procedure ) *9/'/ E-/H/ '2 -1HA ,' E J1/. (F'(1'JF /1 41'J7 G (' 9('1* G' 7HD'F '1 E FJ/, '1 REDUCE 9EDJ'* 1' 'F,'E F/'/, 'JF E3&DG J '2 'HDJF 2JFG G' E-*ED '3* G /1 'JF 5H1* ('J/ *9/'/ -1HA 1' E FJ/. '2 "F,' G /1 *H6J-'* F1E 'A2'1 '4'1G ' (G 'JF E3&DG F4/G '3*, 'JF '-*E'D H,H/ /'1/ G 'JF E-/H/J* E1(H7 (G J version .'5 ('4/. /1 G1 -'D (G*1 '3* E/ F81 1A*G 4H/! (1'J "4F'JJ (' E-J7 F1E 'A2'1 REDUCE H 4JHG J (G '1J1J "F /1 -D E3'&D E*/'HD AJ2J H F3(J* 9'E E+'D 2J1 1' (1'J E-'3(G J F1/G 'J 1EF (' '3*A'/G '2 E*1J 4H'1*3 4JD/ E71- EJ FJE. /1 'JF E+'D '2 1H'(7 2J1 '3*A'/G 1/G 'JE:    'F,'E 'JF E-'3(G (' '3*A'/G '2 (3*G J atensor G (1'J E-'3('* *'F3H1J /1 F1E 'A2'1 REDUCE *9(JG 4/G '3* 3'/G *1 .H'G/ (H/. HDJ (1'J "4F'JJ (' /3*H1G'J '5DJ H 'HDJG J F1E 'A2'1, 'JF E-'3(G 1' (' '3*A'/G '2 "1'JG G' 'F,'E EJ /GJE. '(*/' (G / 2J1 *H,G FJ/: 1: g := mat((1-rs/rr,0,0,0),(0,-1/(1-rs/rr),0,0),(0,0,-rr**2,0),(0,0,0,-(rr**2)*((sin theta)**2))); 2: ig: = 1/g$ 3: array m(3,3),im(3,3),c(3),cr(3,3,3),r(3,3,3,3),ricci(3,3); 4: c(0) := t$ c(1) := rr$ c(2) := theta$ c(3) := phi$ 8: for i := 0:3 do<< for j := 0:3 do<< m(i,j) := g(i+1,j+1); im(i,j) := ig(i+1,j+1) >>;>>; 9: procedure christoffel(a,b,y,x); 9: for i := 0:3 do<< for j := 0:3 do<< for k := 0:3 do<< for s := 0:3 do<< y(i,j,k) := y(i,j,k)+(b(k,s)/2)*(df(a(j,s),x(i))+df(a(i,s),x(j))-df(a(i,j),x(s))) >>;>>;>>;>>; 10: christoffel(m,im,cr,c); 11: for i := 0:3 do<< for j := 0:3 do<< for k := 0:3 do<< for l := 0:3 do<< for n := 0:3 do<< r(i,j,k,l) := r(i,j,k,l)+cr(k,n,i)* cr(l,j,n)-cr(l,n,i)*cr(k,j,n) >>; r(i,j,k,l) := r(i,j,k,l)+df(cr(l,j,i),c(k))-df(cr(k,j,i),c(l)) >>;>>;>>;>>; 12: array r1(3,3,3,3),r2(3,3,3,3); 13: for i := 0:3 do<< for j := 0:3 do<< for k := 0:3 do<< for l := 0:3 do<< for n := 0:3 do<< r1(i,j,k,l) := r1(i,j,k,l) + m(i,n)*r(n,j,k,l) >>;>>;>>;>>;>>; 14: for i := 0:3 do<< for j := 0:3 do<< for k := 0:3 do<< for l := 0:3 do<< for n := 0:3 do<< for s := 0:3 do<< for p := 0:3 do<< r2(i,j,k,l) := r2(i,j,k,l) + im(j,n)*im(k,s)*im(l,p)*r(i,n,s,p) >>;>>;>>;>>;>>;>>;>>; 15: kretchman := 0$ 16: for i := 0:3 do<< for j := 0:3 do<< for k := 0:3 do<< for l := 0:3 do<< kretchman := kretchman + r1(i,j,k,l)*r2(i,j,k,l) >>;>>;>>;>>; 4JHG J ',1'J 'JF / 1' /1 4D 2J1 EJ (JFJ/.   .7 'HD *91JA E'*1J3 g '3* G GE'F E*1J 4H'1*3 4JD/ '3*. ; /1 'F*G'J .7 F*J,G 1' '~ EJ F/. 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