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G}JB|,#P#DE 1 HI BDEHHII 1 B 1 ,^ 1 70,0La- B V,#PH},^ 1 70 0L#L!-* 1P* 1 y0Yj383}mm ݭI}}`8}``|* ? ɛ,`|:-)| / 1L!`DESTINATION CANT BE DOJ}S.SYS0 0H{ 24Δ 28/L!/) 2 Π 2 0 ξK}hAΞB,0 J 1 BDEHI,HÝDE 1HIHIDELSAVE-GIVE L}FILE,START,END(,INIT,RUN)O S0 1`BDEPHI V` S0H 1 L!M}0 0 1L~0`PLEASE TYPE 1 LETTER,0`hhL! 70 1L0L<1 ,;ɛ7,"ɛ:ݦ1ݥN}A"D|ݤD|ȩ:|ȩ|ɛ,,(/+.ީ1 1,ɛ`轤{NAMEO} TOO LONG B VL!` L1I H1EΝDL1|mDiE` V0`8d/8 i:222 1 LP}!ERROR- 138ɛ+,' 20*.. өr2 1``2TOO MANY DIGITSINVALID HEXAQ}DECIMAL PARAMETER800 0 8 00`,0'D800 H,ɛh`2L1NEED D1 THRU D8uR} ECIMAL PARAMETER800 0 8 00`,0'D800 H,ɛh`2L1NEED D1 THRU D8uL 䙣ލȎ!"` !"H h`lDD   T}TRATS:D"NUR 䙣ލȎ!"` !"H h`lDD    @A+EQUATIONNFTOTRYEQUATIOFFFADFBRUNPRINCOUNAS@Se`A3Seu1uSe V}Wc@=0Seu1u 6F&=46F@ W}W=(5HAVE YOU PLACED THE EQUATION AT LINE 1000? (0=N,1=Y)AM"W A ll**THIS PROGRAM WILL FIND A S X}OLUTION TO F(X)=0 GIVEN THE CONTINUOUS FUNCTION F ON THE INTERVAL [A,B]WITH F(A)F(B)<0# A 6-@# A Y}0(PLEASE ENTER A 6-* A06-0(PLEASE ENTER B 6-* A06-H$!8("TH Z}ERE IS NO ROOT IN THIS INTERVAL.;(>(H @&"(PLEASE ENTER THE TOLERANCE&1-(%PLEASE ENTER MAX NUMBER OF ITER [}ATIONS1 5'(DO YOU HAVE A PRINTER?(0=N,1=Y)+5 A /(WORKING ON ITERATION %!/ A / 6- A \}6-6-) A/6-/6-%++&,'@,6-) A/6-" @0,,3a=b ]}=f(p)= 3p=3+!")++&,'@, + A(6-%@2!$!6- ^}! @< 6- @d;,(METHOD FAILED AFTER  ITERATIONS8";e/,3METHOD FAILED AFTER  ITER _}ATIONS/(ROOT="3ROOT=pp(hPLEASE ENTER THE EQUATION USING THE FORMAT: 1000 EQUATI `}ON=YOUR EQUATION. RERUN THE PROGRAM WHEN DONE.&''6-+@$+#@,,%%@$hh(`PLEASE ENTER YOUR E a}QUATION USING THE FORMAT:1000 EQUATION=X. TYPE 'RUN' WHEN FINISHEDD:BISEC$hh(`PLEASE ENTER YOUR E ]^PNTOGPASPRIN@ See@SeeSee@@SeeO5(-HAVE YOU Ec}NTERED THE EQUATION? (0=NO, 1=YES).9E"O A0/+(#DO YOU HAVE A PRINTER? (0=NO,1=YES)/ ddIF G IS CONd}TINUOUS ON THE INTERVAL [A,B] AND G(X) IS WITHIN [A,B] FOR ALL X WITHIN [A,B], THEN **G HAS A FIXED POINT WITHIN [A,B]e} (ENTER TOLERANCE (ENTER MAX ITERATIONS 6-@nn(fENTER YOUR APPROXIMATION OF THE FIXED POINTf} (USE THE BISECTION METHOD TO OBTAIN THE APPROXIMATION).(2 ! A3 6-7 A@< 6-K''(Ng}OW WORKING ON ITERATION .L,"@,3P=G(P)=NO:&,  A P6-%@Z 6-h}d @P--(METHOD FAILED AFTER  ITERATIONS.9"@93METHOD FAILED AFTER  ITERATIONS.i}( FIXED POINT=""@"3 FIXED POINT= ll(dPLEASE ENTER YOUR EQUATION IN TERMS OF X, WHERE GP0=X. FOR j}INSTANCE,IF THE EQUATION WAS X=5X+3, YOU ee(]WOULD WRITE 4000 GP0=5X+3. THIS WILL BE LINE #4000, SO DO NOT FORGET TO Ik}NCLUDE THE 4000. ((( RERUN THE PROGRAM WHEN FINISHED. ((6-+6@,&+@$+#@,,$ D:FIXEDPTl}CLUDE THE 4000. ((( RERUN THE PROGRAM WHEN FINISHED. ((6-+6@,&+@$+#@,,$ D:FIXEDPTnPRINASPTONEQUATIODE@See 9=@3@See 9See @Yn} A`C(0(%DO YOU WANT A PRINT-OUT (0=NO,1=YES)?4@"@C3 Apb(H(=HAVE YOU ENTERED YOUR Eo}QUATION INTO LINE 4000? (0=NO,1=YES)LX"b A0 a(G("@>3*FLOATING POINT ERROR:P2-2P1+P0=0 (APPROX.)( A p0(0(%TURN ON YOUR PR}INTER AND BEGIN AGAIN. D:STEFFEN*FLOATING POINT ERROR:P2-2P1+P0=0 (APPROX.)( A p0(0(%TURN ON YOUR PR(()hzASPRINSTARENPPEQUATIOROOABU@@@ }A 3A 23  A` %%*******************************RRTHIS PROGRAM WILL FIND INTERVALS FOR } WHICH ROOTS OF A GIVEN EQUATION EXISTS.%%******************************* 9@P,!9@P,(L(H(=HAVE }YOU ENTERED THE EQUATION INTO LINE 4000? (0=NO,1=YES)L*" A0,C(0(%DO YOU WANT A PRINT-OUT (0=NO },1=YES)?4@"@C31 A2E(A(6WHERE DO YOU WISH TO START YOUR INTERVAL SEARCH?E<3(/($WHERE } DO YOU WISH TO END THE SEARCH?3>h(h(]LEAST VALUE? (I.E., IF YOU WISH TO EXAMINE [1.001,1.002], .001 WOULD BE THE } LEAST VALUE).?F " ApP%(%( LOOKING AT [,%]d 6- A@n 6-x 6-%6- }A@ 6-$  A @p 0(0(%ENTER YOUR EQUATION USING THE FORMAT: &(&(4000 EQUAT }ION=YOUR EQUATION '('(RERUN THE PROGRAM WHEN DONE. ]]6-+$$$$,&+@$+$$$,,%+@$+$$,,&+@ }$+$,,%+@$,%@$p0(-("TURN ON THE PRINTER AND TRY AGAIN.0X-(-( THERE WHERE  INTERVALS FO }UND.Z6"@63 THERE WHERE  INTERVALS FOUND.[ (b(( INTERVALS:l(v-@(([ }8,,8,] @6-%@J<<([&,&] IS AN INTERVAL WHERE THE EQUATION=0.O ((( }TH"@H3[&,&] IS AN INTERVAL WHERE THE EQUATION=0.^68,-&h 68,-(# @p#((((E }RROR ENCOUNTERED. TRY AGAIN. D:INTERVALERVAL WHERE THE EQUATION=0.^68,-&h 68,-(# @p#((((E /89APRINASBPBQCOUNALPHXDEROOSTREQBBP2A@@A A$}@   @@@ $} A` &&******************************** **HORNER'S METHOD** <<THIS PROGRAM WILL EVALUATE A POLYNOMIAL OF TH$}E FORM: 66PX=A(n)X^n+A(n-1)X^n-1+ ... ...+A1X+A0 qqFOR A CHOSEN X. THE PROGRAM WILL ALSO DETERMINE THE DERIV$}ATIVE P'(X), AND WILL ALSO REDUCE THE POLYNOMIAL.\\ALSO, AN APPROXIMATE ROOT WILL BE DETERMINED USING THE NEWTON- R$}APHSON EQUATION.%%*******************************C(0(%DO YOU WANT A PRINT-OUT (0=NO,1=YES)?4@"@C3 $} Ap1(-("OF WHAT DEGREE IS YOUR POLYNOMIAL?19,9,9,+('(WHAT IS THE VALUE OF YOUR X?+1$}(1(&PLEASE ENTER THE VALUES A0,A1, ... An:- ((A:"$ 68,-((& ( 6-8,$}68,-* 6-8,68&@,-,""-&@@6@.6-+$,%8,68,-0 6-+$,% 68&@$},-1 2'6-+$,%8,'68,-4((P()=66-&8,'8,8 ( (P'()=$}8,:!(!(AN APPROXIMATE ROOT=;&(&(THE EQUATION IS REDUCED TO:<-@6@> ( (+8,$}X^&@@ C"@ ARF A3P()=3P'()=8,3AN APPROXI$}MATE ROOT=##3THE EQUATION IS REDUCED TO:-@6@3+8,X^&@  Ap$}2 (/(!TURN ON THE PRINTER AND TRY AGAIN2X;(;(0ROOT CAN NOT BE DETERMINED-FLOATING POINT ERROR.YD"@D3$}0ROOT CAN NOT BE DETERMINED-FLOATING POINT ERROR.@D:HORNERE DETERMINED-FLOATING POINT ERROR.YD"@D3$H*+/ADPRINXPQCOUNALPHCLEARCLEAX@@(} @  A` %%******************************* ooTHIS PROGRAM WILL (}APPROXIMATE THE VALUE OF AN F(X) OF A POLYNOMIAL GIVEN A SET OF KNOWN POINTS AND AN X 99USING NEVILLE'S ITERATED (} INTERPOLATION%%*******************************C(0(%DO YOU WANT A PRINT-OUT? (0=NO,1=YES)4@"@C3(} Ap4(0(%HOW MANY POINTS DO YOU HAVE TO ENTER?49,9<, - 68,-*-(}"68<,-& * D(@(5WHAT IS THE VALUE OF THE X THAT YOU WANT EVALUATED?D/(/($YOU WILL NOW ENTE(}R YOUR DATA POINTS:-&@ ((X(%@)?"$ 68,-&((F(X(%@))?((}68<,-* 2-@4-@7``68<,-+++&+8&,,,$+8<&@,,,&++&+8,,,$8&(}@<&@,,,'+8,&8&,,9   d-&@i (8, l-&@n-(}&@x (8<,  T(T(*THE APPROXIMATE VALUE OF F(X())=8&@<&@,(}"@ A-&@ 38, -&@-&@ 38<,(}  CC3THE APPROXIMATE VALUE OF F(())=8&@<&@,p0(-("TURN ON THE PRINTER AND TRY(} AGAIN.0XE(B(7FLOATING POINT ERROR: (XP(I)-XP(I-J))=0E D:NEVILLE"TURN ON THE PRINTER AND TRY(aKL ADPRINXPQCOUNALPHCLEARCLEAXUVALUFACTOPARTIAFRAFACTOR@@,} @ ,} A` %%******************************* llTHIS PROGRAM WILL GENERATE DIVIDED-DIFFERE,}NCE COEFFICIENTS OF THE INTERPOLATORY POLYNOMIAL P GIVEN HHX1,X2 ... XN AND F[X(1)],F[X(2)] ... F[X(N,})].%%*******************************C(0(%DO YOU WANT A PRINT-OUT? (0=NO,1=YES)4@"@C3 Ap4,}(0(%HOW MANY POINTS DO YOU HAVE TO ENTER?49,9<, - 68,-*-"68<,-,}& * /(/($YOU WILL NOW ENTER YOUR DATA POINTS:-&@ ((X(%@)?"$ ,}68,-&((F(X(%@))?(68<,-* 2-@4-@7FF68<,-++8<&,}@,,&+8&@<&@,,,'+8,&8&,,9   d-&@i (8, l-&@n,}-&@x (8<,  A(>(3FORWARD DIVIDED-DIFFERENCE COEFFICIENTS:A(-,}&@ (8<, "@ A A-&@ 38, -&,}@-&@ 38<,  303(FORWARD DIVIDED-DIFFERENCE COEFFICIENTS:3(-,}&@ 38<, " Ap0(-("TURN ON THE PRINTER AND TRY AGAIN.0XE(B(7FLOATING POINT ERROR: ,} (XP(I)-XP(I-J))=0Eb@D(@(5WHAT IS THE VALUE OF YOUR X WHICH YOU WANT EVALUATED?DD 6-O:8@,&,}8,,E6-@J6-+&8,,'O 6-T6-8<,^0-@&@06-+$+#,,$,}+8<,,h/!@!6-$+&@,/6-&@j 6-$r 6-%| D(D(THE VALUE OF F(#) IS ,} APPROXIMATLY=77(/(USING NEWTON'S DIV. DIFF. FORMULA).<"@<3THE VALUE OF F() I,}S APPROXIMATLY=@"@@3,(USING NEWTON'S FORWARD DIV. DIFF. FORMULA). D:DIVDIFF3THE VALUE OF F() I,c89  PRINASADTORFEFAROWVAREDRRDIMSU@@@@30}A""?% @uxP ?7D)q) ?7D)q) @ @@@@@@@ @0}#@Y2Va@ @ A`6-@ %%******************************* hhTHIS PROGRAM WILL APPROXIM0}ATE THE INTEGRAL OF A GIVEN FUNCTION FROM A TO B USING ROMBERG'S METHOD OF INTEGRATION. %%**************************0}*****C(0(%DO YOU WANT A PRINT-OUT (0=NO,1=YES)?4@"@C3 Apb(H(=HAVE YOU ENTERED YOUR EQUATIO0}N INTO LINE 4000 (0=NO,1=YES)?LX"b A0I("(WHAT IS THE VALUE OF A?&)(E(WHAT IS THE VALUE OF 0}B?It(O(DHOW MANY ROWS WOULD YOU LIKE GENERATED? (NOTE: A VERY CLOSEt( APPROXIMATION CAN BE ACHIEVED IF0}t+(#YOU SPECIFY A LOT OF ROWS. A LOT OFO(COMPUTER TIME WILL BE REQUIRED,t( HOWEVER, IF YOU SPECIFY TOO MANYr*("ROWS0}. TRY NO MORE THAN 9 AT FIRST.M(IF THE VALUES YOU GET BEGIN TOr( CONVERGE IN AN OBVIOUS WAY, THENK*("YOU KNOW THAT Y0}OU ARE GETTING VERYH(CLOSE TO THE TRUE VALUE.)K(9<, 6-& V 6- A@6-6-) A@/6-0}V68@<@,-+$+%,,'@"%(%(R(1,1)=8@<@,$."@.3R(1,1)=8@<@,(0}-@*%%-@+@#+&@,,,)6-%++&?P,$,# A@)6-- 6-%. /9968@0}<@,-?P$+8@<@,%$,06-2-@4xx68@<,-+++@#+&@,,$+8@0}<&@,,,&8@<&@,,'++@#+&@,,&@,6 <-@>$$(R(,)=8@0}<,@0"@03R(,)=8@<,B F6-'@P-@R#68@<,-8@<,0}# Y Z5(5(APPROXIMATE ANSWER=8@<, UNITS.[A"@3A3APPROXIMATE ANSWER=8@<, UN0}ITS.c l(l(aENTER YOUR EQUATION AS LINE 4000 USINGTHE FORMAT: 4000 EQ=YOUR EQUATION 2(0}/($RERUN THE PROGRAM WHEN YOU ARE DONE.2 6-G:,$p0(-("TURN ON THE PRINTER AND TRY AGAIN.0X#( (FLOAT0}ING POINT ERROR.# D:ROMBERGNE.2 6-G:,$p0(-("TURN ON THE PRINTER AND TRY AGAIN.0X#( (FLOAT0%89\ n PRINXPAHALPHALMZCBDCLEACOUNNUMBEXSPUD@@ A A< Ax A A 4}A, Ah A  A  A   @ @ @P@@WB  A` %%*********4}********************** mmTHIS PROGRAM WILL CONSTRUCT THE CUBIC SPLINE INTERPOLANT S FOR THE FUNCTION F GIVEN X0,X1, ... 4}XN AND ,,F(X0),F(X1), ... F(XN), WHERE F(XI)=AI %%*******************************C(0(%DO YOU WANT A PRINT-OUT (0=4}NO,1=YES)?4@"@C3 Ap4(0(%HOW MANY POINTS DO YOU HAVE TO ENTER?4779,9,9,9,9,4}9,9,9,9,9,k- 68,-/68,->68,-M68,-\68,-k68,-4}C68,-!68,-068,-?68,-C ((((YOU WILL NOW ENTER YOUR DATA:-(-4}&@(-(X()?68,-((F(X())? 68,- (6-&@268,-4}8%@,&8,6 2-@&@468,-@$++++8%@,$8&@,,&+8,$+8%@,&8&@4},,,%+8&@,$8,,,,'+8&@,$8,,,6 8B68,-@-68,-B68,-<4}-@&@>LL68,-+@$+8%@,&8&@,,,&+8&@,$8&@,,@68,-8,'8,B22684},-+8,&+8&@,$8&@,,,'8,D F068,-@!68,-068,-P""-&@64}@R##68,-8,&+8,$8%@,,S666-++8,$+8%@,%+@$8,,,,'@,T''68,-++8%@,&8,4},'8,,&V//68,-+8%@,&8,,'+@$+8,,,W Z-&@d((A()=8,f((X4}()=8,g"&@ Ah((B()=8,j((C()=8,l((D()=8,n ( 4}"-&@3A()=8,3X()=8,"&@ A3B(4})=8,3C()=8,3D()=8, 3 +p/(,(!TURN ON THE PRINTER AND TRY AGAIN/X#(4} (FLOATING POINT ERROR.# D:NATCUBIC)=8, 3 +p/(,(!TURN ON THE PRINTER AND TRY AGAIN/X#(4/  2@@COUN +@AR@(@ (@8} hh161,44,162,58,163,73,164,88,165,99,1669 },111,167,120,168,122,169,123,170,124,171,125,172,124,173,120Hmm174,117,175,111,176,108,177,107,178,109,179,111,180,120,1819 },125,182,127,183,128,184,124,185,122,186,111Rhh187,107,188,90,189,87,190,84,191,80,192,70,193,55,194,40,195,35,196,31,197,9 }28,198,25,199,23,200,22\ii201,23,202,25,203,27,204,29,205,33,206,37,207,42,208,48,209,57,210,67,211,78,212,89,213,125,214,9 }127f 215,128hjj216,129,217,130,218,133,219,127,220,126,221,124,222,122,223,120,224,111,225,103,226,95,227,89,228,84j9 } 229,80pjj230,77,231,73,232,68,233,55,234,45,235,30,236,22,237,18,238,15,239,13,240,10,241,8,242,7,243,6,244,5 # 9}-@@` " """ " """  ,/AP  ',AD@'/ADAP :9}(FIGURE 2:(%RECTANGLES ARE USED TO DETERMINE AREA $$(HIT TO CONTINUE . . . 4F:Ad,"@*Ad9}AU4 A0P 4F:Ad,"@(*AdAU4 A@ A0F "AdAU+"$##9}*************************#6-+# A!( (!(  ( ( (((( 1:INSTRUCTI9}ONS 6:INTEGRAL-(-("2:INTERVALS 7:POLYNOMIAL0(0(%3:ROOT 8:INITIAL VALUE0(0(%9}4:FIXED POINT 9:INTERPOLATION(( 5:DERIVATIVE# ( (((#( SELECT ITEM.4F:Ad,"@1*A9}dAU4 A`9F:Ad,"@0*AdAU9% D:INTERVAL4F:Ad,"@&*AdAU4 9}AB4F:Ad,"@)*AdAU4%D:DER8F:Ad,"@$*AdAU8% D:STEFFEN8F:A9}d,"@'*AdAU8% D:ROMBERG7F:Ad,"@Q*AdAU7%D:HORNER9F:Ad,"@S9}*AdAU9% D:PREDCORR4F:Ad,"@H*AdAU4 AE7F:Ad,"@5*AdAU9}7%D:NEWTON5F:Ad,"@b*AdAU5%D:ZERO8F:Ad,"@*AdAU8% D:NEVILLE9}8F:Ad,"@X*AdAU8% D:DIVDIFF9F:Ad,"@*AdAU9% D:NATCUBIC A@ 9}h##ROOT METHODS*****************i+-@@+( j1-@@1(N:NEWTON-RAPHSON METHO9}Dl)-@@)(S:SECANT METHODn A@IIINTERPOLATION METHODS ******************************************9}****-@@*( -@@(V:NEVILLE'S METHOD##(D:DIVIDED DIFFERENCE METH9}OD(C:CUBIC SPLINE A@p""INSTRUCTIONS****************zAR@* (}*( 9}̠U( (U(GTHIS DISK IS A FUNCTIONAL LIBRARY OF DIFFERENT METHODS OF MATHEMATICALOO(GANALYSES. ALTHOUGH THE9} PROGRAMS DO NOT ALWAYS GIVE ANSWERS, ITKK(CWILL ALLOW THE USER TO GET VERY CLOSE APPROXIMATIONS IN MOST CASES.9 }]](UTHIS PROGRAM, WITH CERTAIN EXCEPTIONS,WILL GIVE VERY GOOD ANSWERS TO GIVEN PROBLEMS.q(q(fTHE INSTRUCTIONS WHICH 9!}FOLLOW WILL GIVE YOU AN OVERVIEW OF THE METHODS THAT ARE AVAILABLE TO YOU.:(:(/HIT TO RETURN TO THE MENU WHENE9"}VERYOU WISH.6-@@"6(HIT TO CONTINUE . . .4F:Ad,"@*AdAU4 A`E4F9#}:Ad,"@(*AdAU4 A@ A`@##INTERVAL*********************$ (}$( INTER9$}VALSY( (Y(KTHIS OPTION WILL ALLOW THE USER TO GENERATE SOME OR ALL OF THE INTERVALS))(!OF A GIVEN EQUATION WHER9%}E F(X)=0.P(P(ETHIS PROGRAM IS USEFULL IN THAT IT WILL GIVE YOU AN IDEA WHERE THE$$(FUNCTION CROSSES THE X-AXIS.9&}b(/($THE INPUT NEEDED TO RUN THE PROGRAM:2(E(1:THE EQUATIONb(2:THE DOMAIN OF INTEREST(3:THE LEAST VALUE9'}G(G(} FUNCTION,,(((F(X+H)-F(X))/H.Q(Q(FAPPROACHES THE TANGENT OF A FUNCTION AT X FOR VERY SMALL H. THE SLOPE033(+9?}OF THIS TANGENT LINE IS THE DERIVATIVEAT X.2/(/($THE INPUT NEEDED TO RUN THE PROGRAM:66-@@"6(HIT 9@} TO CONTINUE . . .84F:Ad,"@*AdAU4 Ab:4F:Ad,"@(*AdAU4 A@<9A} AbB4 (} ((1:THE EQUATION4(2:THE VALUE OF XD6-@@"6(HIT TO CONTINUE . . .F4F:9B}Ad,"@*AdAU4 Ab H4F:Ad,"@(*AdAU4 A@J AbL (}N!(9C}!( INTEGRALPU(U(JTHIS PROGRAM WILL DETERMINE THE INTEGRAL OF AN EQUATION FROM A TO B.RPP(HIN ESS9D}ENCE, THE INTEGRAL OF AN EQUATION FRON A TO B IS SIMPLY THETCC(;THE AREA UNDER THE CURVE OF THE FUNCTION FROM9E} A TO B.V6-@@"6(HIT TO CONTINUE . . .X4F:Ad,"@*AdAU4 Ab@Z4F:A9F}d,"@(*AdAU4 A@\ Ab2` Ab% +(}%AR@dSS(KIN ORDER TO F9G}IND THE AREA UNDER THE GRAPH, THE PROGRAM 'DRAWS' RECTANGLESfPP(HUNDER THE GRAPH. THE AREAS OF EACH RECTANGLE IS COMP9H}UTED, AND THE SUMh>>(6OF THE AREAS IS TAKEN AS THE AREA UNDER THE CURVE.jB-@@"6(HIT TO CONTIN9I}UE . . .B6-@l4F:Ad,"@*AdAU4 Ab`n4F:Ad,"@(*AdAU4 A@9J}p AbRt; +@@10@; Av (}AR@xMM(EYOU MAY HAVE NOTICE9K}D THAT THE RECTANGLES ARE NOT PERFECT. THEz]](UPROGRAM TAKES INTO ACCOUNT THESE DISCREPANCIES WHEN COMPUTING 9L}THE INTEGRAL.|a(/($THE INPUT NEEDED TO RUN THE PROGRAM:2(E(1:THE EQUATIONa(2:THE VALUES OF A AND B~,,($3:9M}THE NUMBER OF ROWS TO BE GENERATEDRR(J (NOTE:THIS IS SORT OF LIKE SETTING A TOLERANCE. THE PROGRAM BUILDS ANNN(F 9N} INTERNAL MATRIX OF VALUES. THE MORE ROWS GENERATED, THE CLOSER THE( THE APPROXIMATION.)6-@@"6(H9O}IT TO CONTINUE . . .4F:Ad,"@*AdAU4 Ab4F:Ad,"@(*AdAU4 9P}A@ Ab (}( POLYNOMIALP(P(ETHIS PROGRAM WILL EVALUATE AN N-TH DEGREE POLYNOMIAL 9Q}AT A GIVEN X.KK(CALSO, THE DERIVATIVE AT X WILL BE COMPUTED, AN APPROXIMATE ROOTQQ(IWILL BE CALCULATED, AND THE P9R}OLYNOMIALWILL BE REDUCED TO AN ORDER OF N-1./(/($THE INPUT NEEDED TO RUN THE PROGRAM:^(&(1:THE COEFFICIENTS A0 T9S}O AN;(2:THE VALUE OF X^(3:THE DEGREE OF THE POLYNOMIAL6-@@"6(HIT TO CONTINUE . . .4F:A9T}d,"@*AdAU4 Ac 4F:Ad,"@(*AdAU4 A@ Ac (} ( 9U} INITIAL VALUET(T(ITHIS PROGRAM WILL COMPUTE AN INITAL VALUE FUNCTION FOR A GIVEN EQUATION(IN Y AND T 9V}FROM A TO B./(/($THE INPUT NEEDED TO RUN THE PROGRAM:m($(1:THE EQUATION IN Y AND T<(2:THE INITIAL VALUEJ( 39W}:A AND Bm(4:THE NUMBER OF ITERATIONS (N)V(V(KTHE NUMBER OF ITERATIONS INPUTED WILL DETERMINE THE SUBINTERVALS EVALUA9X}TED.4'(THE SIZE OF THE SUBINTERVALS IS4((B-A)/N.6-@@"6(HIT TO CONTINUE . . .4F:Ad9Y},"@*AdAU4 AcP4F:Ad,"@(*AdAU4 A@ Ac8 (} ( 9Z} INTERPOLATIONJ(J(?THERE ARE THREE TYPES OF INTERPOLATIONINCLUDED IN THIS SECTION:+(+( 1:NEVILLE'S INTERPOLAT9[}ION METHODU(U(JGIVEN A SET OF EVENLY SPACED X VALUES (EG., X0=1, X1=.8, X2=.6, ETC.), ANDOO(GTHE CORRESPONDING Y VA9\}LUES, THIS PROGRAM WILL DETERMINE THE PROPERTT(LVALUE OF F(XN) GIVEN AN XN SUCH THAT XN LIES BETWEEN YOUR SET OF KN9]}OWN X'S.P,($THE INPUT NEEDED TO RUN THE PROGRAM:/(P(1:THE NUMBER OF KNOWN POINTSS(2:YOUR X AND Y VALUESS(13:T9^}HE VALUE OF X THAT YOU WANT EVALUATED6-@@"6(HIT TO CONTINUE . . .4F:Ad,"@9_}*AdAU4 Ac4F:Ad,"@(*AdAU4 A@ Acp (}66(.2:DIVIDED DIFFER9`}ENCE INTERPOLATION METHODP(P(ETHIS METHOD IS SIMILAR TO NEVILLE'S METHOD OF INTERPOLATION, AND ITSS(KREQUIRE9a}S THE SAME INPUT. AS A MATTER OF FACT, IT WORKS MUCH BETTER AS LONGPP(HAS THE VALUE OF X THAT YOU WANT EVALUATED I9b}S CLOSE TO YOUR TOP X'SKK(C(I.E., YOUR X'S ARE ARRANGED IN A STACK: X0, X1, X2, X3, ETC.).HH(@IF THIS REQUIREMENT9c} CAN NOT BE MET, THEN USE NEVILLE'S METHOD.d(d(YTHIS PROGRAM WILL ALSO GENERATE FORWARD DIVIDED DIFFERENCE 9d} COEFFICIENTS.6-@@"6(HIT TO CONTINUE . . .4F:Ad,"@*AdAU4 Ad9e}4F:Ad,"@(*AdAU4 A@ Ac  (} (3:NATURAL CUBIC SPLINER(R(GTHIS PROG9f}RAM WILL GERNERATE THE NATURAL CUBIC SPLINE COEFFICIENTSOO(GGIVEN POINTS OF A GRAPH OF A CONTINUOUS FUNCTI9g}ON (THAT IS, THE``(XFUNCTION IS EVERYWHERE DIFFER- ENTIABLE). THE EQUATION OF THE SPLINE IS GIVEN BY:_(<(1S(9q}B%DOS SYSB*)DUP SYSBSAUTORUN SYSB UBISEC B bFIXEDPT BmNEWTON B|SECANT BSTEFFEN BINTERVAL BHORNER B NEVILLE BDIVDIFF BROMBERG BNATCUBIC BSTART BPREDCORR B DER BZERO X)=Si(X)=Ai+Bi(X-Xi)+Ci(X-Xi)^2+ +Di(X-Xi)^3_(ON THE INTERVAL [X(i),X(i+1)].R(R(GAi, Bi, Ci, Di, AND Xi ARE THE 9r} NUMBERS GENERATED IN THE PROGRAM.R(R(GIF PROPERLY USED, THE CUBIC SPLINE EQUATIONS THAT YOU CONSTRUCT WILL-9s}-(%ENABLE YOU TO GENERATE SMOOTH GRAPHS.6-@@"6(HIT TO CONTINUE . . .4F:Ad,"@*A9t}dAU4 Ad@ 4F:Ad,"@(*AdAU4 A@" Ad0( (}*PP(HIT IS IMPORTANT THAT 9u}THE FUNCTION IS CONTINUOUS. IF IT IS NOT, THEN YOU,SS(KCOULD SIMPLY GENERATE A SERIES OF SPLINES - EACH SERIES FOR EA9v}CH DOMAIN.))(!WHERE THE FUNCTION IS CONTINUOUS.0/(/($THE INPUT NEEDED TO RUN THE PROGRAM:2H(!(1:THE NUMBER OF P9w}OINTSH("2:ALL CORRESPONDING X AND Y POINTS4S(S(HTHIS PROGRAM ASSUMES SOME KNOWLEDGE OF CUBIC SPLINE INTERPOLATION. 9x}IF6LL(DYOU WISH TO LEARN MORE, THEN I WOULD SUGGEST THAT YOU GET A BOOK ON866(.NUMERICAL ANALYSIS FROM YOUR LOCAL L9y}IBRARY.<6-@@"6(HIT TO CONTINUE . . .>4F:Ad,"@*AdAU4 Adp@4F:Ad9z},"@(*AdAU4 A@B AdbF (}H%%( GENERAL CONSIDERATIONSJO(O(DWHEN TYPING IN 9{}EQUATIONS, BE SURE THAT THEY ARE WELL-NESTED. FORLa%(INSTANCE, INSTEAD OF WRITING:((@((3*(X^4))+(5*(X^2))C(^(9|}USE THIS FORM INSTEAD:a(N((3*X*X*X*X)+(5*X*X)PU(U(JTHE PROGRAM WILL RUN MUCH FASTER, AND THE APPROXIMATIONS GENER9}}ATED WILL BER&&(MUCH CLOSER TO THE TRUE VALUE.TP(P(EBE SURE TO NOTE THE TOLERANCE IN THE DERIVATIVE PROGRAM. THE PR9~}OGRAMVSS(KPERFORMS ITERATIONS SUCH THAT THE DENOMINATOR GETS SMALLER AND SMALLER.Z6-@@"6(HIT 9}TO CONTINUE . . .\4F:Ad,"@*AdAU4 Ad^4F:Ad,"@(*AdAU4 A@` 9} Adc (}dQQ(ITHIS CAN SOMETIMES RESULT IN A DIVERGING SEQUENCE, AND THUS A VERYf(BAD APPROXIMATION!9}hn(n(cI WOULD SAY THAT A TOLERANCE OF <=0.0001 IS GOOD. A TOLERANCE OF 3, HOWEVER, REALLY STINKS!jQ(Q(FI HA9}VE NOT SET ANY PROTECTIONS ON THE PROGRAMS. THIS WILL ALLOW YOU TOl=:(2MAKE MODIFICATIONS TO THE PROGRAMS IF YOU WISH TO.9}=(nQQ(ISOME OF THESE PROGRAMS PRESUPPOSES SOME MATHEMATICAL KNOWLEDGE. AGAIN,pDD( 5 ?c! u>Vq>SI> C?b6d?aP?`ac@6xy(?c! u@=} A` %%******************************* jjTHIS PROGRAM WILL EMPLOY THE ADAMS-BASHFORTH METHOD AS PREDICTOR =} AND THE ADAMS-MOULTON METHOD AS hhCORRECTOR IN ORDER TO APPROXIMATE THE INITIAL VALUE PROBLEM AT N+1 EQUALLY SPACED NU=}MBERS IN THE 33INTERVAL [A,B] GIVEN Y'=F(T,Y) AND Y(A)=ALPHA%%*******************************C(0(%DO YOU WANT A =}PRINT-OUT (0=NO,1=YES)?4@"@C3 Apb(H(=HAVE YOU ENTERED YOUR EQUATION INTO LINE 4000 (0=NO,1=YES=})?LX"b A0)((ENTER A(%(ENTER B)-(( ENTER ALPHA()(ENTER N-6-+&=},'9,9,!68,-!68,-"("(8,,8,-@@76-8&@=},%6-8&@,/ A@76-$ O 6-8&@,%+'@,=6-8&@,%+'@,G A@O6-$!O 6-8=}&@,%+'@,=6-8&@,%+'@,G A@O6-$";6-8&@,%)6-8&@,%3 A@;6-=}$#;;68,-8&@,%++%@$%@$%,'@,$68,-%+$,&(8,,8,'"@38,,=}8,( 2-@46-%+$,516-8@,!6-8@,+ A@16-616-8@,!6-8@=},+ A@16-716-8@,!6-8@,+ A@16-9FF6-8@,%++'@,$++@#$,&+@$,%+=}@$,,,<%6-%+$,6- A@%6->HH6-8@,%++'@$,$++@ $,%+@$,&+@$,%,,@ =}(,A"@3,F-@H+68,-8%@,+68,-8%@,J P!68@,-=}!68@,-Z _ ( (APPROXIMATE ANSWER=`)"@)3APPROXIMATE ANSWER=c B(B(7ENTER YOUR EQU=}ATION INTO LINE 4000 USING THE FORMAT: ( (4000 EQ=YOUR EQUATION 2(/($RERUN THE PROGRAM WHEN YOU ARE DONE.2=}6-6%%@$p0(-("TURN ON THE PRINTER AND TRY AGAIN.0X((FLOATING POINT ERROR D:PREDCORR<z!"$ASPRINTOXEDECHECKTAN@=q>yqC? )@@RFc@Qx@Qx @ A}A6GA - @@  A`9A, %%******************************* ffTHIS PROGRAM WILL DETERMINEA} THE DERIVATIVE OF AN EQUATION AT X GIVEN THE EQUATION IN TERMS OF X. ppA TOLERANCE OF .0000001 WILL TRY TO BE ACHIEVED.A} IF IT IS NOT POSSIBLE TO ACHIEVE THIS TOLERANCE, IITHEN THE PROGRAM WILL GIVE THE DERIVATIVE AT X AS WELL AS POSSIA}BLE.%%*******************************b(H(=HAVE YOU ENTERED YOUR EQUATION INTO LINE 4000 (0=NO,1=YES)?LX"A}b A0C(0(%DO YOU WANT A PRINT-OUT (0=NO,1=YES)?4@"@C3 Ap6-@+('(WHAT A}IS THE VALUE OF YOUR X?+ 6-% A@6-  6-& A@6-"##6-+@'+@$,,$+&,$ ( (A}%"@3( 68,-2!@ @p46-%@66-'@8 @0FR7O:8,&8&@A},,!O:8&@,&8&@,,H6-8&@,R @M,"O:8,&8&@,, <, @N$P"("(F'(A} ) IS APPROX.=R+"@+3F'( ) IS APPROX.=UZ$$6-O:8&@,&8&@,,\(( TOLERANCEA}=^ "@ 3 TOLERANCE=b @ B(B(7ENTER YOUR EQUATION INTO LINE 4000 USING THE FORMAT: ( (A}4000 EQ=YOUR EQUATION 2(/($RERUN THE PROGRAM WHEN YOU ARE DONE.26-G:,$+@$+$$,,$p1(.(#TURN ONA} YOUR PRINTER AND TRY AGAIN.1X)(&(ERROR ENCOUNTED. TRY AGAIN.) D:DER@$+$$,,$p1(.(#TURN ON@V/0  PRINASPPTONQEQUATIOQAPPROXFFF@SeGSeedS=@U@SeedE}S# r r SeecA N Thuu@DSP?QB9 A`E( (2($DO YOU WANT A PRINE}T-OUT(0=NO,1=YES)?6B"@E3 Apb(H(=HAVE YOU ENTERED YOUR EQUATION INTO LINE 4000 (0=NO,1=YES)?LE}X"b A0"((ENTER THE TOLERANCE" A 8()( ,(/(2(8(E}-()(ENTER MAX NUMBER OF ITERATIONS-%%*******************************RRTHIS PROGRAM WILL APPROXIMATE A ROOTE} OF AN EQUATION USING THE SECANT METHOD.mmGIVEN TWO INITIAL APPROXIMATIONS (P0 & P1), THIS METHOD PROVIDES RAPID CONVERE}GENCE TO THE ROOT. THIS__METHOD DOES NOT REQUIRE THE DERIVATIVE OF THE EQUATION AS THE NEWTON-RAPHSON METHOD DOES.%%E}*******************************6-@ AP 6- A@! 6-# 6- A@6-( ! E}A26-&++$+&,,'+&,,3#(#(NOW WORKING ITERATION 4((P=5"@3P=<O:E}&,  A F6-%@P+ 6-6-6-6-% A@+6-Z @1,,(METHOD FAILED AFTER  ITE}ERATIONS9"@93METHOD FAILED AFTER  ITERATIONS. ( ((ROOT="@3ROOT=E} B(B(7ENTER YOUR EQUATION INTO LINE 4000 USING THE FORMAT: )(&(4000 EQUATION=YOUR EQUATION)( '('(RERUE}N THE PROGRAM WHEN DONE. ''6-+@$+#@,,%%@$(( (((Q1-Q0=0: OVERFLOW ERROR.3"@E}3 33Q1-Q0=0: OVERFLOW ERROR.( A p0(-("TURN ON THE PRINTER AND TRY AGAIN.0X)(&(ERROR E}ENCOUNTED. TRY AGAIN.)@9A3,All**THIS PROGRAM WILL FIND A SOLUTION TO F(X)=0 GIVEN THE CONTINUOUS FUNCTION E}F ON THE INTERVAL [A,B]BWITH F(A)F(B)<0C# Ap6-@# A@D3((PLEASE ENTER A#6-- A@E}36-E3((PLEASE ENTER B#6-- A@36-FK$!(;("THERE IS NO ROOT IN THIS INTERVAL.>(A(K E}AJ"("(WORKING ON ITERATION K/ 6- A@6-6-) A@/6-T/6-%++&,'@,6-) A@E}/6-W 68,-Y" A0Z,,3a=b=f(p)=[ 3p=]3^1'")+E}+&,'@, ?1 Ah6-%@r!$!6-! A| 6- A(#-($(FIRST APPROXIE}MATION=8,-6-8,)# A*#>(-(SECOND APPROXIMATION=8&@,>6-8&@,,#-"@-3FIRST APPE}ROXIMATION=8,.#6"@63SECOND APPROXIMATION=8&@,2# @ #>(-(SECOND APPROXIMATION=8,&>E}>6-8,&>#6"@63SECOND APPROXIMATION=8,&># @ D:ZEROPPROXIMATION=8,&>Df