From: eldorado@ecn.purdue.edu (Dave Jansen) Newsgroups: comp.sources.hp48 Subject: v04i008: laplace_dj - Laplace & Inverse Laplace Transforms v1.0, Part01/01 Date: 29 Feb 92 20:06:24 GMT Followup-To: comp.sys.hp48 Organization: Univ. of North Carolina @ Wilmington Checksum: 2466395008 (verify with brik -cv) Submitted-by: Dave Jansen Posting-number: Volume 4, Issue 8 Archive-name: laplace_dj/part01 BEGIN_DOC laplace.doc These are some public domain routines to work in conjunction with Wayne Scott's polynomial routines. The routines included will find any given level of a Pascal triangle, convert the output of the PF (partial fraction routine) complex linear terms to real quadratic terms, take this output and find Inverse Laplace Transforms, and find Laplace and Inverse Laplace Transforms of equations. PASCALTRI - Finds a level of Pascal's triangle example : Input 1: 4 Output 1: { 1 4 6 4 1 } CONV - When using the partial fraction program in Wayne Scott's Poly routines, quadratic terms in the denominator must be split into their corresponding linear terms (this is done with the RT or Root program). This is necessary to find the partial fraction, but afterwards it is often important to work with real coefficients. This routine will convert all the complex terms in a poly list to the original quadratic with real coefficients. example : Polynomial Form Actual Equation Input 2: { 3 -2.5 1.25 1.25 } 3 -2.5 1.25 1.25 1: { -4 -4 (-2,2) (-2,-2) } ----- + ----- + -------- + -------- (S+4) (S+4) (S+2-2i) (S+2+2i) Output 2: { 3 -2.5 { 2.5 5 } } 3 -2.5 (2.5S + 5) 1: { 4 4 { 4 8 } } ----- + ----- + -------------- (S+4) (S+4) (S^2 + 4S + 8) Notice that the values in the denominator are no longer the roots but instead the coefficients of a polynomial. For this reason, the values change sign. Also, the nested list is a linear term in the numerator and a quadratic in the denominator. The coefficient in front of the S^2 term is always one and so is not included in the list. L2EQ - This routine converts the output list from the CONV routine above and returns the Inverse Laplace of the polynomial equation. example : Input 2: { 3 -2.5 { 2.5 5 } } 3 -2.5 (2.5S + 5) 1: { 4 4 { 4 8 } } ----- + ----- + -------------- (S+4) (S+4) (S^2 + 4S + 8) Output 1: '2.5*EXP(-2*T)*COS(sqrt(4)*T)*u(T)+(-(2.5*EXP(-4*T)*u(T)) +3/(2-1)!*T^(2-1)*EXP(-4*T)*u(T))' or after using COLCT in the Algebra directory: '3*EXP(-(4*T))*u(T)*T+2.5*EXP(-(2*T))*COS(2*T)*u(T)-2.5*EXP(-(4*T))*u(T)' Notice the routine will handle real repeated roots but so far will not convert repeated complex roots. L-> - Takes the Inverse Laplace Transform of an equation of any of the forms listed below. It is rather picky about signs, parenthesis, the order of subterms, and the presence of leading coefficients. The L2EQ routine will automatically take care of these little annoyances. For example, in the 1/(S+a) term, the sign must be a + and not a - sign in your equation, or else it will not transform it. About the parenthesis, ((5*S)/(S^2+4)) is not the same as 5*S/(S^2+4) and so will not transform. The ordering within terms is very important. S*5/(S^2+4) will not transform correctly to 5*COS(2*T) but instead to S*5/2*SIN(2*T). Also a leading coefficient must be present in front of the term or it will not transform: S/(S^2+4) will not transform; instead use 1*S/(S^2+4). Time and frequency shifting is supported. 1/(S+a) will transform to 1*EXP(-a*T)*u(T) and 1/S*EXP(a*S) will convert to 1*u(T-a). In the use of the time shift, the EXP(a*S) term must be at the end of the term and the time shift will show up in the u(T-a) term but not in the transformed equation. This means that 1/s^2*EXP(T-2) returns 1*T*u(T-2) but should be 1*(T-2)*u(T-2). For the Dirac Delta function, d(T) was chosen instead of the delta symbol, since that is already used for differentiation. The following is a list of terms this routine recognizes: '1/S' '1/(S+a)' '1/S^2' '1/S^n' '1/(S+a)^n' 'S/(S^2+w^2)' '1/(S^2+w^2)' '(S+a)/((S+a)^2+w^2)' '1/((S+a)^2+w^2)' '(S^2-w^2)/(S^2+w^2)^2' '2wS/(S^2+w^2)^2' '(S*COS(h)-w*SIN(h))/(S^2+w^2)' '(S*SIN(h)+w*COS(h))/(S^2+w^2)' '1/((S+a)*(S+b))' ->L - Takes the Laplace Transform of an equation. Below is a listing of the forms the routine will recognize. Notice the sign conventions, parenthetic ordering, requirement of a leading coefficient, and subterm ordering as in the Laplace Transform routine. 'd(T)' 'u(T)' 'EXP(-a*T)*u(T)' 'r(T)' 'T^n*u(T)' '(T-a)^n*u(T-a)' 'T^n*EXP(-a*T)*u(T)' 'COS(w*T)*u(T)' 'SIN(w*T)*u(T)' 'EXP(-a*T)*COS(w*T)*u(T)' 'EXP(-a*T)*SIN(w*T)*u(T)' 'T*COS(w*T)*u(T)' 'T*SIN(w*T)*u(T)' 'COS(w*T+h)*u(T)' 'SIN(w*T+h)*u(T)' '(EXP(-a*T)-EXP(-b*T))*u(T)' Some tips and suggestions I have are to save the denominator before doing a partial fraction expansion (PF) to use for the CONV routine. This way you do not have to re-enter it later. The routines that find the Laplace and Inverse Laplace Transforms take a little while (atleast 15 seconds depending on the complexity of the equation being transformed). For this reason, L2EQ is slowed down since it calls the Inverse Laplace Transform routine. The reason I did not combine the partial fractions and Inverse Laplace Transform into one routine is to allow the user to edit the output from the roots (ex: 2.00000000001 should be changed to 2.0). Remember that repeated roots MUST be adjacent in the list for the routines to work correctly. Wayne's partial fraction (PF) routine requires this, as does the L2EQ routine. As you will find out, it will be useful to use the COLCT function in the Algebra menu to reduce the equations. Two examples follow. example 1 S + 3 ------------------- S^3 + 5S^2 + 8S + 4 2: { 1 3 } 1: { 1 5 8 4 } RT 4: { 1 3 } 3: -.999999999991 2: -1.99999799053 1: -2.00000200949 round off and put in list observing correct root order ( 3 ->LIST ) 2: { 1 3 } 1: { -1 -2 -2 } copy denominator ( DUP ROT ROT ) 3: { -1 -2 -2 } 2: { 1 3 } 1: { -1 -2 -2 } PF 2: { -1 -2 -2 } 1: { 2 -1 -2 } SWAP 2: { 2 -1 -2 } 1: { -1 -2 -2 } CONV 2: { 2 -1 -2 } 1: { 1 2 2 } L2EQ 1: '-(2*EXP(-2*T)*u(T))+(-1/(2-1)!*T^(2-1)*EXP(-2*T)*u(T)+2*EXP(-1*T)*u(T))' COLCT 1: '-(EXP(-(2*T))*u(T)*T)+2*EXP(-T)*u(T)-2*EXP(-(2*T))*u(T)' example 2 8(S^2 + 4S + 3) ------------------------ (S + 4)^2 (S^2 + 4S + 8) multiplies out to 2: { 8 32 24 } 1: { 1 12 56 128 128 } RT 2: { 8 32 24 } 1: { -4 -4 (-2,2) (-2,-2) } PF 2: { 3 -2.5 1.25 1.25 } 1: { -4 -4 (-2,2) (-2,-2) } CONV 2: { 3 -2.5 { 2.5 5 } } 1: { 4 4 { 4 8 } } L2EQ 1: '2.5*EXP(-2*T)*COS(sqrt(4)*T)*u(T)+(-(2.5*EXP(-4*T)*u(T)) +3/(2-1)!*T^(2-1)*EXP(-4*T)*u(T))' COLCT 1: '3*EXP(-(4*T))*u(T)*T+2.5*EXP(-(2*T))*COS(2*T)*u(T)-2.5*EXP(-(4*T))*u(T)' As you can see, these will help will Laplace Transforms but will not do them for you. A student learning Laplace Transforms will still need to know them just to use these routines. I have tested them for some time and know not of any/many bugs but I am aware of its limitations, as anyone who uses these. Please send any questions, comments and bug reports. Dave ________________________________________________________________________________ David Jansen | INTERNET: eldorado@en.ecn.purdue.edu Electrical Engineering | BITNET: eldorado%ea.ecn.purdue.edu@purccvm Purdue University | UUCP: {purdue, pur-ee}!en.ecn.purdue.edu!eldorado END_DOC BEGIN_RPL laplace.rpl %%HP: T(3)A(D)F(.); DIR L2EQ \<< 0 0 0 0 0 0 0 0 0 0 \-> Nlist Dlist a b c d e f g h I dups \<< Dlist SIZE 1 FOR I IF I Dlist SIZE \=/ THEN IF Dlist I GET Dlist I 1 + GET SAME THEN 'dups' 1 STO+ ELSE 1 'dups' STO END ELSE 1 'dups' STO END Dlist I GET IF DUP SIZE 2 == THEN OBJ\-> DROP 'd' STO 'c' STO Nlist I GET OBJ\-> DROP 'b' STO 'a' STO IF c 0 == THEN IF dups 2 == THEN 'S^2' d d * + 2 ^ DUP 'S^2' d d * - SWAP / 2 d * 'S' * ROT / + ELSE 'S^2' d d * + DUP a 'S' * SWAP / b ROT / + END ELSE 1 c d 3 \->LIST RT DROP C\->R 'f' STO 'e' STO a b 2 \->LIST 1 e NEG 2 \->LIST PDIV OBJ\-> DROP 'h' STO OBJ\-> DROP 'g' STO 'S' e NEG IF DUP TYPE 0 == e 0 < AND THEN 0 R\->C END + DUP g SWAP * SWAP 2 ^ f f * + DUP ROT ROT / SWAP h SWAP / + END ELSE IF DUP 0 == THEN IF dups 1 > THEN DROP Nlist I GET 'S' dups ^ / ELSE DROP Nlist I GET 'S' / END ELSE IF dups 1 > THEN 'S' SWAP IF DUP DUP TYPE 0 == SWAP 0 < AND THEN 0 R\->C END + dups ^ Nlist I GET SWAP / ELSE 'S' SWAP IF DUP DUP TYPE 0 == SWAP 0 < AND THEN 0 R\->C END + Nlist I GET SWAP / END END END -1 STEP IF Dlist SIZE 1 > THEN 2 Dlist SIZE START + NEXT END L\-> IF DEPTH 1 > THEN IF SWAP DUP TYPE 5 == THEN OBJ\-> DROP 'd(T)' * + END END \>> \>> CONV \<< 0 0 0 0 0 0 \-> numer denom count len a b r1 r2 \<< numer SIZE 'count' STO WHILE count 1 > REPEAT denom count GET 'r2' STO denom count 1 - GET 'r1' STO IF r1 IM 0 \=/ r2 IM 0 \=/ AND r1 CONJ r2 == AND THEN numer count 1 - GET 'a' STO numer count GET 'b' STO denom count 1 - r1 r2 + NEG r1 r2 * 2 \->LIST PUT OBJ\-> DUP 'len' STO count - 1 + ROLL DROP len 1 - \->LIST 'denom' STO numer count 1 - a b + a r2 * b r1 * + NEG 2 \->LIST PUT OBJ\-> DUP 'len' STO count - 1 + ROLL DROP len 1 - \->LIST 'numer' STO 'count' -2 STO+ ELSE denom count DUP2 GET NEG PUT 'denom' STO 'count' -1 STO+ END END IF count 1 == THEN denom 1 DUP2 GET NEG PUT 'denom' STO END numer denom \>> \>> \->L \<< { '&k*(T-&a)^ &n*u(T-&a)' '&k*&n! /S^(&n+1)*EXP(-&a*S )' } \|vMATCH DROP { '&k*u(T-&a)' '&k*u( T)*EXP(-&a*S)' } \|vMATCH DROP { '&k*r (T-&a)' '&k*r(T)* EXP(-&a*S)' } \|vMATCH DROP { '&k*d (T-&a)' '&k*d(T)* EXP(-&a*S)' } \|vMATCH DROP { '&k*d (T)' &k } \|vMATCH DROP { '&k*r(T)' ' &k/S^2' } \|vMATCH DROP { '&k*T^&n*u(T )' '&k*&n!/S^(&n+1) ' } \|vMATCH DROP { ' &k*T^&n*EXP(&a*T)*u (T)' '&k*&n!/(S-&a) ^(&n+1)' } \|vMATCH DROP { '&k*EXP(&a*T )*COS(&\Gw*T)*u(T)' ' &k*(S-&a)/((S-&a)^2 +&\Gw^2)' } \|vMATCH DROP { '&k*EXP(&a*T )*SIN(&\Gw*T)*u(T)' ' &k*&\Gw/((S-&a)^2+&\Gw^ 2)' } \|vMATCH DROP { '&k*T*COS(&\Gw*T)*u(T )' '&k*(S^2-&\Gw^2)/( S^2+&\Gw^2)^2' } \|vMATCH DROP { '&k*T *SIN(&\Gw*T)*u(T)' ' &k/(2*&\Gw)*S/(S^2+&\Gw )^2' } \|vMATCH DROP { '&k*COS(&\Gw*T+&\Gh)* u(T)' '&k*(S*COS(&\Gh )-&\Gw*SIN(&\Gh))/(S^2+ &\Gw^2)' } \|vMATCH DROP { '&k*SIN(&\Gw*T +&\Gh)*u(T)' '&k*(S* SIN(&\Gh)+&\Gw*COS(&\Gh)) /(S^2+&\Gw^2)' } \|vMATCH DROP { '&k* COS(&\Gw*T)*u(T)' '&k *S/(S^2+&\Gw^2)' } \|vMATCH DROP { '&k* SIN(&\Gw*T)*u(T)' '&k *&\Gw/(S^2+&\Gw^2)' } \|vMATCH DROP { '&k*( EXP(&\Ga*T)-EXP(&\Gg*T) )*u(T)' '&k/((S-&\Ga) *(S-&\Gg))' } \|vMATCH DROP { '&k*EXP(&a*T )*u(T)' '&k/(S-&a)' } \|vMATCH DROP { '&k *u(T)' '&k/S' } \|vMATCH DROP \>> L\-> \<< { '&k/S' '&k* u(T)' } \|vMATCH DROP { '&k/(S+&a)' '&k* EXP(-&a*T)*u(T)' } \|vMATCH DROP { '&k/S ^2' '&k*r(T)' } \|vMATCH DROP { '&k/S ^&n' '&k/(&n-1)!*T^ (&n-1)*u(T)' } \|vMATCH DROP { '&k/( S+&a)^&n' '&k/(&n-1 )!*T^(&n-1)*EXP(-&a *T)*u(T)' } \|vMATCH DROP { '&k/S^&n*EXP (&a*S)' '&k/(&n-1)! *(T-&a)^(&n-1)*u(T- &a)' } \|vMATCH DROP { '&k*(S+&a)/((S+&a )^2+&\Gw)' '&k*EXP(- &a*T)*COS(\v/&\Gw*T)*u( T)' } \|vMATCH DROP { '&k/((S+&a)^2+&\Gw)' '&k/\v/&\Gw*EXP(-&a*T)* SIN(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k*( S^2-&\Gw)/(S^2+&\Gw)^2' '&k*T*COS(\v/&\Gw*T)*u( T)' } \|vMATCH DROP { '&k*S/(S^2+&\Gw)^2' ' &k/(2*\v/&\Gw)*T*SIN(\v/ &\Gw*T)*u(T)' } \|vMATCH DROP { '&k*( S*COS(&\Gh)-&\Gw*SIN(&\Gh ))/(S^2+&\Gr)' '&k* COS(&\Gw*T+&\Gh)*u(T)' } \|vMATCH DROP { '&k *(S*SIN(&\Gh)+&\Gw*COS( &\Gh))/(S^2+&\Gr)' '&k* SIN(&\Gw*T+&\Gh)*u(T)' } \|vMATCH DROP { '&k *S/(S^2+&\Gw)' '&k* COS(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k/( S^2+&\Gw)' '&k/\v/&\Gw* SIN(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k/( (S+&\Ga)*(S+&\Gg))' '&k /(&\Gg-&\Ga)*(EXP(-&\Ga*T )-EXP(-&\Gg*T))*u(T)' } \|vMATCH DROP { '&k *&f(T)*EXP(&a*S)' ' &k*&f(T+&a)' } \|vMATCH DROP { '&k* EXP(&a*S)' '&k*d(T+ &a)' } \|vMATCH DROP \>> PASCALTRI \<< \-> n \<< IF n 0 == THEN { 1 } ELSE IF n 1 == THEN { 1 1 } ELSE { 1 1 } 1 n 1 - START { 1 1 } PMUL NEXT END END \>> \>> END END_RPL BEGIN_ASC laplace.asc %%HP: T(3)A(D)F(.); "69A20FF75362000000900514353414C445259490D9D20E16321C432D6E2010E6 E16323CE22D6E2010E64B2A2279E1AFE2247A209C2A2B21305BF22D9D203CE22 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M*RIRZ:'O(ITM,.PB;2Y`0%8'-Y\BG2V`O1]M+E#@Q)8V1]?F`@%) MQM=!92-(+A`PE6392\!HK,5#](BLQ4/LBG2TP[")M+D!`5@RA M[R*=+4!E(T@N$#"59R.]^S'L(H?[<;@?ALM!*RIRZ=&['[2BXKL>@^>A[R*= M+4`K*I['L1(#U2]RMAIM+D!`5@![Z M+M+9`K2BXGD<*S%0_2)GJ]'F`@5.;&ES=&TN$)!D?!V]^U'P&BLQ4/TB*S%0 M_2(K,5#](H:C`C@CPR[2Y@(%1&QI\BG2W@+2IM+E!`Q)8V M1X>;'`,Q\^!O1^X*H#D`@%4N"K0Y@(!9"LQD/\#0/:Q$@/NK7&V 3&BLQ4/TB*S%0_2+^-9)C(RLQ`"LQ ` end END_UU From: eldorado@en.ecn.purdue.edu (Dave Jansen) Newsgroups: comp.sys.hp48 Subject: Laplace & Inverse Laplace Transforms Date: 16 Feb 92 05:38:04 GMT Organization: Purdue University Engineering Computer Network These are some public domain routines to work in conjunction with Wayne Scott's polynomial routines. The routines included will find any given level of a Pascal triangle, convert the output of the PF (partial fraction routine) complex linear terms to real quadratic terms, take this output and find Inverse Laplace Transforms, and find Laplace and Inverse Laplace Transforms of equations. PASCALTRI - Finds a level of Pascal's triangle example : Input 1: 4 Output 1: { 1 4 6 4 1 } CONV - When using the partial fraction program in Wayne Scott's Poly routines, quadratic terms in the denominator must be split into their corresponding linear terms (this is done with the RT or Root program). This is necessary to find the partial fraction, but afterwards it is often important to work with real coefficients. This routine will convert all the complex terms in a poly list to the original quadratic with real coefficients. example : Polynomial Form Actual Equation Input 2: { 3 -2.5 1.25 1.25 } 3 -2.5 1.25 1.25 1: { -4 -4 (-2,2) (-2,-2) } ----- + ----- + -------- + -------- (S+4) (S+4) (S+2-2i) (S+2+2i) Output 2: { 3 -2.5 { 2.5 5 } } 3 -2.5 (2.5S + 5) 1: { 4 4 { 4 8 } } ----- + ----- + -------------- (S+4) (S+4) (S^2 + 4S + 8) Notice that the values in the denominator are no longer the roots but instead the coefficients of a polynomial. For this reason, the values change sign. Also, the nested list is a linear term in the numerator and a quadratic in the denominator. The coefficient in front of the S^2 term is always one and so is not included in the list. L2EQ - This routine converts the output list from the CONV routine above and returns the Inverse Laplace of the polynomial equation. example : Input 2: { 3 -2.5 { 2.5 5 } } 3 -2.5 (2.5S + 5) 1: { 4 4 { 4 8 } } ----- + ----- + -------------- (S+4) (S+4) (S^2 + 4S + 8) Output 1: '2.5*EXP(-2*T)*COS(sqrt(4)*T)*u(T)+(-(2.5*EXP(-4*T)*u(T)) +3/(2-1)!*T^(2-1)*EXP(-4*T)*u(T))' or after using COLCT in the Algebra directory: '3*EXP(-(4*T))*u(T)*T+2.5*EXP(-(2*T))*COS(2*T)*u(T)-2.5*EXP(-(4*T))*u(T)' Notice the routine will handle real repeated roots but so far will not convert repeated complex roots. L-> - Takes the Inverse Laplace Transform of an equation of any of the forms listed below. It is rather picky about signs, parenthesis, the order of subterms, and the presence of leading coefficients. The L2EQ routine will automatically take care of these little annoyances. For example, in the 1/(S+a) term, the sign must be a + and not a - sign in your equation, or else it will not transform it. About the parenthesis, ((5*S)/(S^2+4)) is not the same as 5*S/(S^2+4) and so will not transform. The ordering within terms is very important. S*5/(S^2+4) will not transform correctly to 5*COS(2*T) but instead to S*5/2*SIN(2*T). Also a leading coefficient must be present in front of the term or it will not transform: S/(S^2+4) will not transform; instead use 1*S/(S^2+4). Time and frequency shifting is supported. 1/(S+a) will transform to 1*EXP(-a*T)*u(T) and 1/S*EXP(a*S) will convert to 1*u(T-a). In the use of the time shift, the EXP(a*S) term must be at the end of the term and the time shift will show up in the u(T-a) term but not in the transformed equation. This means that 1/s^2*EXP(T-2) returns 1*T*u(T-2) but should be 1*(T-2)*u(T-2). For the Dirac Delta function, d(T) was chosen instead of the delta symbol, since that is already used for differentiation. The following is a list of terms this routine recognizes: '1/S' '1/(S+a)' '1/S^2' '1/S^n' '1/(S+a)^n' 'S/(S^2+w^2)' '1/(S^2+w^2)' '(S+a)/((S+a)^2+w^2)' '1/((S+a)^2+w^2)' '(S^2-w^2)/(S^2+w^2)^2' '2wS/(S^2+w^2)^2' '(S*COS(h)-w*SIN(h))/(S^2+w^2)' '(S*SIN(h)+w*COS(h))/(S^2+w^2)' '1/((S+a)*(S+b))' ->L - Takes the Laplace Transform of an equation. Below is a listing of the forms the routine will recognize. Notice the sign conventions, parenthetic ordering, requirement of a leading coefficient, and subterm ordering as in the Laplace Transform routine. 'd(T)' 'u(T)' 'EXP(-a*T)*u(T)' 'r(T)' 'T^n*u(T)' '(T-a)^n*u(T-a)' 'T^n*EXP(-a*T)*u(T)' 'COS(w*T)*u(T)' 'SIN(w*T)*u(T)' 'EXP(-a*T)*COS(w*T)*u(T)' 'EXP(-a*T)*SIN(w*T)*u(T)' 'T*COS(w*T)*u(T)' 'T*SIN(w*T)*u(T)' 'COS(w*T+h)*u(T)' 'SIN(w*T+h)*u(T)' '(EXP(-a*T)-EXP(-b*T))*u(T)' Some tips and suggestions I have are to save the denominator before doing a partial fraction expansion (PF) to use for the CONV routine. This way you do not have to re-enter it later. The routines that find the Laplace and Inverse Laplace Transforms take a little while (atleast 15 seconds depending on the complexity of the equation being transformed). For this reason, L2EQ is slowed down since it calls the Inverse Laplace Transform routine. The reason I did not combine the partial fractions and Inverse Laplace Transform into one routine is to allow the user to edit the output from the roots (ex: 2.00000000001 should be changed to 2.0). Remember that repeated roots MUST be adjacent in the list for the routines to work correctly. Wayne's partial fraction (PF) routine requires this, as does the L2EQ routine. As you will find out, it will be useful to use the COLCT function in the Algebra menu to reduce the equations. Two examples follow. example 1 S + 3 ------------------- S^3 + 5S^2 + 8S + 4 2: { 1 3 } 1: { 1 5 8 4 } RT 4: { 1 3 } 3: -.999999999991 2: -1.99999799053 1: -2.00000200949 round off and put in list observing correct root order ( 3 ->LIST ) 2: { 1 3 } 1: { -1 -2 -2 } copy denominator ( DUP ROT ROT ) 3: { -1 -2 -2 } 2: { 1 3 } 1: { -1 -2 -2 } PF 2: { -1 -2 -2 } 1: { 2 -1 -2 } SWAP 2: { 2 -1 -2 } 1: { -1 -2 -2 } CONV 2: { 2 -1 -2 } 1: { 1 2 2 } L2EQ 1: '-(2*EXP(-2*T)*u(T))+(-1/(2-1)!*T^(2-1)*EXP(-2*T)*u(T)+2*EXP(-1*T)*u(T))' COLCT 1: '-(EXP(-(2*T))*u(T)*T)+2*EXP(-T)*u(T)-2*EXP(-(2*T))*u(T)' example 2 8(S^2 + 4S + 3) ------------------------ (S + 4)^2 (S^2 + 4S + 8) multiplies out to 2: { 8 32 24 } 1: { 1 12 56 128 128 } RT 2: { 8 32 24 } 1: { -4 -4 (-2,2) (-2,-2) } PF 2: { 3 -2.5 1.25 1.25 } 1: { -4 -4 (-2,2) (-2,-2) } CONV 2: { 3 -2.5 { 2.5 5 } } 1: { 4 4 { 4 8 } } L2EQ 1: '2.5*EXP(-2*T)*COS(sqrt(4)*T)*u(T)+(-(2.5*EXP(-4*T)*u(T)) +3/(2-1)!*T^(2-1)*EXP(-4*T)*u(T))' COLCT 1: '3*EXP(-(4*T))*u(T)*T+2.5*EXP(-(2*T))*COS(2*T)*u(T)-2.5*EXP(-(4*T))*u(T)' As you can see, these will help will Laplace Transforms but will not do them for you. A student learning Laplace Transforms will still need to know them just to use these routines. I have tested them for some time and know not of any/many bugs but I am aware of its limitations, as anyone who uses these. Please send any questions, comments and bug reports. Cut Here 8<--------------------------------------------------------------------- %%HP: T(3)A(D)F(.); DIR L2EQ \<< 0 0 0 0 0 0 0 0 0 0 \-> Nlist Dlist a b c d e f g h I dups \<< Dlist SIZE 1 FOR I IF I Dlist SIZE \=/ THEN IF Dlist I GET Dlist I 1 + GET SAME THEN 'dups' 1 STO+ ELSE 1 'dups' STO END ELSE 1 'dups' STO END Dlist I GET IF DUP SIZE 2 == THEN OBJ\-> DROP 'd' STO 'c' STO Nlist I GET OBJ\-> DROP 'b' STO 'a' STO IF c 0 SAME THEN IF dups 2 == THEN 'S^2' d d * + 2 ^ DUP 'S^2' d d * - SWAP / 2 d * 'S' * ROT / + ELSE 'S^2' d d * + DUP a 'S' * SWAP / b ROT / + END ELSE 1 c d 3 \->LIST RT DROP C\->R 'f' STO 'e' STO a b 2 \->LIST 1 e NEG 2 \->LIST PDIV OBJ\-> DROP 'h' STO OBJ\-> DROP 'g' STO 'S' e - DUP g SWAP * SWAP 2 ^ f f * + DUP ROT ROT / SWAP h SWAP / + END ELSE IF DUP 0 == THEN IF dups 1 > THEN DROP Nlist I GET 'S' dups ^ / ELSE DROP Nlist I GET 'S' / END ELSE IF dups 1 > THEN 'S' SWAP + dups ^ Nlist I GET SWAP / ELSE 'S' SWAP + Nlist I GET SWAP / END END END -1 STEP IF Dlist SIZE 1 > THEN 2 Dlist SIZE START + NEXT END L\-> IF DEPTH 1 > THEN IF SWAP DUP TYPE 5 SAME THEN OBJ\-> DROP 'd(T)' * + END END \>> \>> CONV \<< 0 0 0 0 0 0 \-> numer denom count len a b r1 r2 \<< numer SIZE 'count' STO WHILE count 1 > REPEAT denom count GET 'r2' STO denom count 1 - GET 'r1' STO IF r1 IM 0 \=/ r2 IM 0 \=/ AND r1 CONJ r2 == AND THEN numer count 1 - GET 'a' STO numer count GET 'b' STO denom count 1 - r1 r2 + NEG r1 r2 * 2 \->LIST PUT OBJ\-> DUP 'len' STO count - 1 + ROLL DROP len 1 - \->LIST 'denom' STO numer count 1 - a b + a r2 * b r1 * + NEG 2 \->LIST PUT OBJ\-> DUP 'len' STO count - 1 + ROLL DROP len 1 - \->LIST 'numer' STO 'count' -2 STO+ ELSE denom count DUP2 GET NEG PUT 'denom' STO 'count' -1 STO+ END END IF count 1 == THEN denom 1 DUP2 GET NEG PUT 'denom' STO END numer denom \>> \>> \->L \<< { '&k*(T-&a)^ &n*u(T-&a)' '&k*&n! /S^(&n+1)*EXP(-&a*S )' } \|vMATCH DROP { '&k*u(T-&a)' '&k*u( T)*EXP(-&a*S)' } \|vMATCH DROP { '&k*r (T-&a)' '&k*r(T)* EXP(-&a*S)' } \|vMATCH DROP { '&k*d (T-&a)' '&k*d(T)* EXP(-&a*S)' } \|vMATCH DROP { '&k*d (T)' &k } \|vMATCH DROP { '&k*r(T)' ' &k/S^2' } \|vMATCH DROP { '&k*T^&n*u(T )' '&k*&n!/S^(&n+1) ' } \|vMATCH DROP { ' &k*T^&n*EXP(&a*T)*u (T)' '&k*&n!/(S-&a) ^(&n+1)' } \|vMATCH DROP { '&k*EXP(&a*T )*COS(&\Gw*T)*u(T)' ' &k*(S-&a)/((S-&a)^2 +&\Gw^2)' } \|vMATCH DROP { '&k*EXP(&a*T )*SIN(&\Gw*T)*u(T)' ' &k*&\Gw/((S-&a)^2+&\Gw^ 2)' } \|vMATCH DROP { '&k*T*COS(&\Gw*T)*u(T )' '&k*(S^2-&\Gw^2)/( S^2+&\Gw^2)^2' } \|vMATCH DROP { '&k*T *SIN(&\Gw*T)*u(T)' ' &k/(2*&\Gw)*S/(S^2+&\Gw )^2' } \|vMATCH DROP { '&k*COS(&\Gw*T+&\Gh)* u(T)' '&k*(S*COS(&\Gh )-&\Gw*SIN(&\Gh))/(S^2+ &\Gw^2)' } \|vMATCH DROP { '&k*SIN(&\Gw*T +&\Gh)*u(T)' '&k*(S* SIN(&\Gh)+&\Gw*COS(&\Gh)) /(S^2+&\Gw^2)' } \|vMATCH DROP { '&k* COS(&\Gw*T)*u(T)' '&k *S/(S^2+&\Gw^2)' } \|vMATCH DROP { '&k* SIN(&\Gw*T)*u(T)' '&k *&\Gw/(S^2+&\Gw^2)' } \|vMATCH DROP { '&k*( EXP(&\Ga*T)-EXP(&\Gg*T) )*u(T)' '&k/((S-&\Ga) *(S-&\Gg))' } \|vMATCH DROP { '&k*EXP(&a*T )*u(T)' '&k/(S-&a)' } \|vMATCH DROP { '&k *u(T)' '&k/S' } \|vMATCH DROP \>> L\-> \<< { '&k/S' '&k* u(T)' } \|vMATCH DROP { '&k/(S+&a)' '&k* EXP(-&a*T)*u(T)' } \|vMATCH DROP { '&k/S ^2' '&k*r(T)' } \|vMATCH DROP { '&k/S ^&n' '&k/(&n-1)!*T^ (&n-1)*u(T)' } \|vMATCH DROP { '&k/( S+&a)^&n' '&k/(&n-1 )!*T^(&n-1)*EXP(-&a *T)*u(T)' } \|vMATCH DROP { '&k/S^&n*EXP (&a*S)' '&k/(&n-1)! *(T-&a)^(&n-1)*u(T- &a)' } \|vMATCH DROP { '&k*(S+&a)/((S+&a )^2+&\Gw)' '&k*EXP(- &a*T)*COS(\v/&\Gw*T)*u( T)' } \|vMATCH DROP { '&k/((S+&a)^2+&\Gw)' '&k/\v/&\Gw*EXP(-&a*T)* SIN(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k*( S^2-&\Gw)/(S^2+&\Gw)^2' '&k*T*COS(\v/&\Gw*T)*u( T)' } \|vMATCH DROP { '&k*S/(S^2+&\Gw)^2' ' &k/(2*\v/&\Gw)*T*SIN(\v/ &\Gw*T)*u(T)' } \|vMATCH DROP { '&k*( S*COS(&\Gh)-&\Gw*SIN(&\Gh ))/(S^2+&\Gr)' '&k* COS(&\Gw*T+&\Gh)*u(T)' } \|vMATCH DROP { '&k *(S*SIN(&\Gh)+&\Gw*COS( &\Gh))/(S^2+&\Gr)' '&k* SIN(&\Gw*T+&\Gh)*u(T)' } \|vMATCH DROP { '&k *S/(S^2+&\Gw)' '&k* COS(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k/( S^2+&\Gw)' '&k/\v/&\Gw* SIN(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k/( (S+&\Ga)*(S+&\Gg))' '&k /(&\Gg-&\Ga)*(EXP(-&\Ga*T )-EXP(-&\Gg*T))*u(T)' } \|vMATCH DROP { '&k *&f(T)*EXP(&a*S)' ' &k*&f(T+&a)' } \|vMATCH DROP { '&k* EXP(&a*S)' '&k*d(T+ &a)' } \|vMATCH DROP \>> PASCALTRI \<< \-> n \<< IF n 0 == THEN { 1 } ELSE IF n 1 == THEN { 1 1 } ELSE { 1 1 } 1 n 1 - START { 1 1 } PMUL NEXT END END \>> \>> END -- The weather is here, wish you were beautiful! ________________________________________________________________________________ David Jansen | INTERNET: eldorado@en.ecn.purdue.edu From: eldorado@ecn.purdue.edu (Dave Jansen) Newsgroups: comp.sources.hp48 Subject: v04i039: laplace_dj - Laplace & Inverse Laplace Transforms v2.0, Part01/01 Date: 29 Mar 92 19:05:29 GMT Followup-To: comp.sys.hp48 Organization: Univ. of North Carolina @ Wilmington Checksum: 3257894907 (verify with brik -cv) Submitted-by: Dave Jansen Posting-number: Volume 4, Issue 39 Archive-name: laplace_dj/part01 Supersedes: laplace_dj: Volume 4, Issue 8 BEGIN_DOC laplace.doc These are some public domain routines to work in conjunction with Wayne Scott's polynomial routines which are included in this posting. The routines included will convert the output of the PF (partial fraction routine) complex linear terms to real quadratic terms, take this output and find Inverse Laplace Transforms, and find Laplace and Inverse Laplace Transforms of equations of one sided transforms. ->QUAD - When using the partial fraction program in Wayne Scott's Poly routines, quadratic terms in the denominator must be split into their corresponding linear terms (this is done with the RT or Root program). This is necessary to find the partial fraction, but afterwards it is often important to work with real coefficients. This routine will convert all the complex terms in a poly list to the original quadratic with real coefficients. example : Polynomial Form Actual Equation Input 2: { 3 -2.5 1.25 1.25 } 3 -2.5 1.25 1.25 1: { -4 -4 (-2,2) (-2,-2) } ----- + ----- + -------- + -------- (s+4) (s+4) (s+2-2i) (s+2+2i) Output 2: { 3 -2.5 { 2.5 5 } } 3 -2.5 (2.5s + 5) 1: { 4 4 { 4 8 } } ----- + ----- + -------------- (s+4) (s+4) (s^2 + 4s + 8) Notice that the values in the denominator are no longer the roots but instead the coefficients of a polynomial. For this reason, the values of the denominator change sign. Also, the nested list is a linear term in the numerator and a quadratic in the denominator. The coefficient in front of the s^2 term is always 1 and so is not included in the list. L2EQ - This routine will find the partial fraction expansion, convert the output list using the ->QUAD routine above and finally returns the Inverse Laplace of the polynomial equation. example : Polynomial Form Input 2: { 8 32 24 } numerator 1: { -4 -4 (-2,2) (-2,-2) } roots of denominator Output 1: '2.5*EXP(-(2*T))*COS(2*T)*u(T)-2.5*EXP(-(4*T))*u(T) +3*T*EXP(-(4*T))*u(T))' Notice the routine will handle real repeated roots but I am not sure if the repeated complex root case is correct. L-> - Takes the Inverse Laplace Transform of an equation of any of the forms listed below. It is rather picky about signs, parenthesis, the order of subterms, and the presence of leading coefficients. The L2EQ routine will automatically take care of these little annoyances. For example, in the 1/(S+a) term, the sign must be a + and not a - sign in your equation, or else it will not transform it. About the parenthesis, ((5*S)/(S^2+4)) is not the same as 5*S/(S^2+4) and so will not transform. The ordering within terms is very important. S*5/(S^2+4) will not transform correctly to 5*COS(2*T) but instead to S*5/2*SIN(2*T). Also a leading coefficient must be present in front of the term or it will not transform: S/(S^2+4) will not transform; instead use 1*S/(S^2+4). Time and frequency shifting is supported. 1/(S+a) will transform to 1*EXP(-a*T)*u(T) and 1/S*EXP(a*S) will convert to 1*u(T-a). In the use of the time shift, the EXP(a*S) term must be at the end of the term and the time shift will show up in the u(T-a) term but not in the transformed equation. This means that 1/s^2*EXP(T-2) returns 1*T*u(T-2) but should be 1*(T-2)*u(T-2). For the Dirac Delta function, d(T) was chosen instead of the delta symbol, since that is already used for differentiation. The following is a list of terms this routine recognizes: '1/S' '1/(S+a)' '1/S^2' '1/S^n' '1/(S+a)^n' 'S/(S^2+w^2)' '1/(S^2+w^2)' '(S+a)/((S+a)^2+w^2)' '1/((S+a)^2+w^2)' '(S^2-w^2)/(S^2+w^2)^2' '2wS/(S^2+w^2)^2' '(S*COS(h)-w*SIN(h))/(S^2+w^2)' '(S*SIN(h)+w*COS(h))/(S^2+w^2)' '1/((S+a)*(S+b))' ->L - Takes the Laplace Transform of an equation. Below is a listing of the forms the routine will recognize. Notice the sign conventions, parenthetic ordering, requirement of a leading coefficient, and subterm ordering as in the Laplace Transform routine. 'd(T)' 'u(T)' 'EXP(-a*T)*u(T)' 'r(T)' 'T^n*u(T)' '(T-a)^n*u(T-a)' 'T^n*EXP(-a*T)*u(T)' 'COS(w*T)*u(T)' 'SIN(w*T)*u(T)' 'EXP(-a*T)*COS(w*T)*u(T)' 'EXP(-a*T)*SIN(w*T)*u(T)' 'T*COS(w*T)*u(T)' 'T*SIN(w*T)*u(T)' 'COS(w*T+h)*u(T)' 'SIN(w*T+h)*u(T)' '(EXP(-a*T)-EXP(-b*T))*u(T)' The routines that find the Laplace and Inverse Laplace Transforms take a little while (atleast 15 seconds depending on the complexity of the equation being transformed). It is strongly adviseable to round the roots produced by the RT routine (ex: 2.00000000011 should be changed to 2.0). Remember that repeated roots MUST be adjacent in the list for the routines to work correctly. However since Wayne's partial fraction (PF) routine requires this also, this is not usually a problem. It may be necessary to use the COLCT function in the Algebra menu to reduce the equations. Two examples follow. example 1 s + 3 ------------------- s^3 + 5s^2 + 8s + 4 2: { 1 3 } 1: { 1 5 8 4 } RT 4: { 1 3 } 3: -.999999999991 2: -1.99999799053 1: -2.00000200949 3 ->LIST 2: { 1 3 } 1: { -1.999999999991 -1.99999799053 -2.00000200949 } edit 2: { 1 3 } 1: { -1 -2 -2 } L2EQ 1: '-(2*EXP(-(2*T))*u(T))-T*EXP(-(2*T))*u(T)+2*EXP(-T)*u(T))' example 2 8(s^2 + 4s + 3) ------------------------ (s + 4)^2 (s^2 + 4s + 8) multiplies out to 2: { 8 32 24 } 1: { 1 12 56 128 128 } RT 2: { 8 32 24 } 1: { -4 -4 (-2,2) (-2,-2) } L2EQ 1: '2.5*EXP(-(2*T))*COS(2*T)*u(T)-2.5*EXP(-(4*T))*u(T) +3*T*EXP(-(4*T))*u(T))' I have tested these for some time and know not of any/many bugs but I am aware of its limitations, as anyone who uses these should also be. Please send any questions, comments and bug reports. END_DOC BEGIN_SRC laplace.rpl %%HP: T(3)A(D)F(.); DIR L2EQ \<< DUP ROT ROT PF SWAP \->QUAD 0 0 0 0 0 0 \-> numer denom count a b r1 r2 dups \<< denom SIZE 1 FOR count IF count denom SIZE < THEN IF denom count GETI ROT ROT GET == THEN 'dups' 1 STO+ ELSE 1 'dups' STO END ELSE 1 'dups' STO END denom count GET IF DUP TYPE 5 == THEN OBJ\-> DROP 'r2' STO 'r1' STO IF r1 THEN denom count GET { 1 0 0 } PADD RT DROP C\->R 'r2' STO NEG 'r1' STO numer count GET 1 r1 2 \->LIST PDIV OBJ\-> DROP 'b' STO OBJ\-> DROP 'a' STO 'a*EXP (-r1*T)*COS(r2*T)*u (T)+b/r2*EXP(-r1*T) *SIN(r2*T)*u(T)' EVAL ELSE numer count GET OBJ\-> DROP 'b' STO 'a' STO r2 \v/ 'r2' STO IF dups 1 > THEN 'a*T*COS(r2*T)*u(T) +b/(2*r2)*T*SIN(r2* T)*u(T)' EVAL ELSE 'a*COS(r2*T)*u(T)+b /r2*SIN(r2*T)*u(T)' EVAL END END ELSE 'r1' STO numer count GET 'a' STO IF r1 THEN IF dups 1 > THEN 'a/(dups-1)!*T^( dups-1)*EXP(-r1*T)* u(T)' EVAL ELSE 'a*EXP(-r1*T)*u(T)' EVAL END ELSE IF dups 1 > THEN 'a/(dups-1)!*T^( dups-1)*u(T)' EVAL ELSE 'a*u(T)' EVAL END END END IF count denom SIZE < THEN + END -1 STEP \>> \>> \->QUAD \<< DUP SIZE 0 0 0 0 \<< IF DUP IM NOT THEN RE END \>> \-> numer denom count a b r1 r2 reduce \<< WHILE count 1 > REPEAT denom count 1 - GETI 'r1' STO GET 'r2' STO IF r1 IM 0 \=/ r1 CONJ r2 == AND THEN numer count 1 - GETI 'a' STO GET 'b' STO denom count 1 - r1 r2 + NEG reduce EVAL r1 r2 * reduce EVAL 2 \->LIST PUT OBJ\-> DUP count - 2 + ROLL DROP 1 - \->LIST 'denom' STO numer count 1 - a b + reduce EVAL a r2 * b r1 * + NEG reduce EVAL 2 \->LIST PUT OBJ\-> DUP count - 2 + ROLL DROP 1 - \->LIST 'numer' STO 'count' -2 STO+ ELSE denom count DUP2 GET NEG PUT 'denom' STO numer count DUP2 GET reduce EVAL PUT 'numer' STO 'count' -1 STO+ END END numer denom IF count THEN SWAP 1 DUP2 GET reduce EVAL PUT SWAP 1 DUP2 GET NEG PUT END \>> \>> \->L \<< { '&k*(T-&a)^ &n*u(T-&a)' '&k*&n! /S^(&n+1)*EXP(-&a*S )' } \|vMATCH DROP { '&k*u(T-&a)' '&k*u( T)*EXP(-&a*S)' } \|vMATCH DROP { '&k*r (T-&a)' '&k*r(T)* EXP(-&a*S)' } \|vMATCH DROP { '&k*d (T-&a)' '&k*d(T)* EXP(-&a*S)' } \|vMATCH DROP { '&k*d (T)' &k } \|vMATCH DROP { '&k*r(T)' ' &k/S^2' } \|vMATCH DROP { '&k*T^&n*u(T )' '&k*&n!/S^(&n+1) ' } \|vMATCH DROP { ' &k*T^&n*EXP(&a*T)*u (T)' '&k*&n!/(S-&a) ^(&n+1)' } \|vMATCH DROP { '&k*EXP(&a*T )*COS(&\Gw*T)*u(T)' ' &k*(S-&a)/((S-&a)^2 +&\Gw^2)' } \|vMATCH DROP { '&k*EXP(&a*T )*SIN(&\Gw*T)*u(T)' ' &k*&\Gw/((S-&a)^2+&\Gw^ 2)' } \|vMATCH DROP { '&k*T*COS(&\Gw*T)*u(T )' '&k*(S^2-&\Gw^2)/( S^2+&\Gw^2)^2' } \|vMATCH DROP { '&k*T *SIN(&\Gw*T)*u(T)' ' &k/(2*&\Gw)*S/(S^2+&\Gw )^2' } \|vMATCH DROP { '&k*COS(&\Gw*T+&\Gh)* u(T)' '&k*(S*COS(&\Gh )-&\Gw*SIN(&\Gh))/(S^2+ &\Gw^2)' } \|vMATCH DROP { '&k*SIN(&\Gw*T +&\Gh)*u(T)' '&k*(S* SIN(&\Gh)+&\Gw*COS(&\Gh)) /(S^2+&\Gw^2)' } \|vMATCH DROP { '&k* COS(&\Gw*T)*u(T)' '&k *S/(S^2+&\Gw^2)' } \|vMATCH DROP { '&k* SIN(&\Gw*T)*u(T)' '&k *&\Gw/(S^2+&\Gw^2)' } \|vMATCH DROP { '&k*( EXP(&\Ga*T)-EXP(&\Gg*T) )*u(T)' '&k/((S-&\Ga) *(S-&\Gg))' } \|vMATCH DROP { '&k*EXP(&a*T )*u(T)' '&k/(S-&a)' } \|vMATCH DROP { '&k *u(T)' '&k/S' } \|vMATCH DROP \>> L\-> \<< { '&k/S' '&k* u(T)' } \|vMATCH DROP { '&k/(S+&a)' '&k* EXP(-&a*T)*u(T)' } \|vMATCH DROP { '&k/S ^2' '&k*r(T)' } \|vMATCH DROP { '&k/S ^&n' '&k/(&n-1)!*T^ (&n-1)*u(T)' } \|vMATCH DROP { '&k/( S+&a)^&n' '&k/(&n-1 )!*T^(&n-1)*EXP(-&a *T)*u(T)' } \|vMATCH DROP { '&k/S^&n*EXP (&a*S)' '&k/(&n-1)! *(T-&a)^(&n-1)*u(T- &a)' } \|vMATCH DROP { '&k*(S+&a)/((S+&a )^2+&\Gw)' '&k*EXP(- &a*T)*COS(\v/&\Gw*T)*u( T)' } \|vMATCH DROP { '&k/((S+&a)^2+&\Gw)' '&k/\v/&\Gw*EXP(-&a*T)* SIN(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k*( S^2-&\Gw)/(S^2+&\Gw)^2' '&k*T*COS(\v/&\Gw*T)*u( T)' } \|vMATCH DROP { '&k*S/(S^2+&\Gw)^2' ' &k/(2*\v/&\Gw)*T*SIN(\v/ &\Gw*T)*u(T)' } \|vMATCH DROP { '&k*( S*COS(&\Gh)-&\Gw*SIN(&\Gh ))/(S^2+&\Gr)' '&k* COS(&\Gw*T+&\Gh)*u(T)' } \|vMATCH DROP { '&k *(S*SIN(&\Gh)+&\Gw*COS( &\Gh))/(S^2+&\Gr)' '&k* SIN(&\Gw*T+&\Gh)*u(T)' } \|vMATCH DROP { '&k *S/(S^2+&\Gw)' '&k* COS(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k/( S^2+&\Gw)' '&k/\v/&\Gw* SIN(\v/&\Gw*T)*u(T)' } \|vMATCH DROP { '&k/( (S+&\Ga)*(S+&\Gg))' '&k /(&\Gg-&\Ga)*(EXP(-&\Ga*T )-EXP(-&\Gg*T))*u(T)' } \|vMATCH DROP { '&k *&f(T)*EXP(&a*S)' ' &k*&f(T+&a)' } \|vMATCH DROP { '&k* EXP(&a*S)' '&k*d(T+ &a)' } \|vMATCH DROP \>> PF \<< MAXR { } \-> Z P OLD LAST \<< 1 P SIZE FOR I P I GET \-> p1 \<< IF p1 OLD \=/ THEN Z p1 EVPLY 1 P SIZE FOR J IF P J GET P I GET \=/ THEN p1 P J GET - / END NEXT p1 'OLD' STO { } 'LAST' STO ELSE IF { } LAST SAME THEN 1 { } 1 P SIZE FOR K P K GET IF DUP p1 == THEN DROP ELSE + END NEXT IRT Z SWAP ELSE LAST OBJ\-> DROP END RDER DUP2 5 PICK 1 + 3 ROLLD 3 \->LIST 'LAST' STO p1 EVPLY SWAP p1 EVPLY SWAP / SWAP ! / END \>> NEXT P SIZE \->LIST \>> \>> RT \<< TRIM DUP SIZE \-> n \<< IF n 3 > THEN DUP BAIRS SWAP OVER PDIV DROP \-> A B \<< A RT B RT \>> ELSE IF n 2 > THEN QUD ELSE LIST\-> DROP NEG SWAP / END END \>> \>> IRT \<< OBJ\-> \-> n \<< IF n 0 > THEN 1 n START n ROLL { 1 } SWAP NEG + NEXT ELSE { 1 } END IF n 1 > THEN 2 n START PMUL NEXT END \>> \>> FCTP \<< IF DUP SIZE 3 > THEN DUP BAIRS SWAP OVER PDIV DROP FCTP END \>> LST \<< \->LIST \>> PMUL \<< DUP2 SIZE SWAP SIZE \-> X Y ny nx \<< 1 nx ny + 1 - FOR I 0 NEXT 1 nx FOR I 1 ny FOR J I J + 1 - ROLL X I GET Y J GET * + I J + 1 - ROLLD NEXT NEXT { } 1 nx ny + 1 - START SWAP + NEXT \>> \>> PDIV \<< DUP SIZE 3 ROLLD OBJ\-> \->ARRY SWAP OBJ\-> \->ARRY \-> c b a \<< a b IF c 1 SAME THEN OBJ\-> DROP / OBJ\-> 1 GET \->LIST { 0 } ELSE WHILE OVER SIZE 1 GET c \>= REPEAT DIVV END DROP \-> d \<< a SIZE 1 GET c 1 - - IF DUP NOT THEN 1 END \->LIST d OBJ\-> OBJ\-> DROP \->LIST \>> END \>> \>> PADD \<< DUP2 SIZE SWAP SIZE \-> A B nB nA \<< A L\178A B L\178A IF nA nB < THEN SWAP END IF nA nB \=/ THEN 1 nA nB - ABS START 0 NEXT END nA nB - ABS 1 + ROLL OBJ\-> 1 GET nA nB - ABS + \->ARRY + L\178A \>> \>> TRIM \<< OBJ\-> \-> n \<< n WHILE ROLL DUP ABS NOT n 1 - AND REPEAT DROP 'n' DECR END n ROLLD n \->LIST \>> \>> PASCALTRI \<< \-> n \<< IF n 0 == THEN { 1 } ELSE IF n 1 == THEN { 1 1 } ELSE { 1 1 } 1 n 1 - START { 1 1 } PMUL NEXT END END \>> \>> RDER \<< \-> F G \<< G F PDER PMUL G PDER { -1 } PMUL F PMUL PADD G G PMUL \>> \>> PDER \<< OBJ\-> \-> n \<< 1 n FOR i n ROLL n i - * NEXT DROP IF n 1 == THEN { 0 } ELSE n 1 - \->LIST END \>> \>> L\178A \<< IF DUP TYPE 5 == THEN OBJ\-> \->ARRY ELSE OBJ\-> 1 GET \->LIST END \>> EVPLY \<< OVER IF DUP TYPE 5 == THEN SIZE ELSE SIZE 1 GET END \-> a r n \<< a 1 GET IF n 1 > THEN 2 n FOR i r * a i GET + NEXT END \>> \>> COEF \<< \-> E n \<< 0 n FOR I 0 'X' STO E EVAL 'X' PURGE E 'X' \.d 'E' STO I ! / NEXT 2 n 1 + FOR I I ROLL NEXT n 1 + \->LIST \>> \>> DIVV \<< DUP 1 GET \-> a b c \<< a 1 GET c / DUP b * a SIZE RDM a SWAP - OBJ\-> 1 GETI 1 - PUT \->ARRY SWAP DROP b \>> \>> QUD \<< LIST\-> \->ARRY DUP 1 GET / ARRY\-> DROP ROT DROP SWAP 2 / NEG DUP SQ ROT - \v/ DUP2 + 3 ROLLD - \>> BAIRS \<< OBJ\-> 1 1 \-> n R S \<< DO 0 n 1 + PICK 0 0 0 4 PICK 5 n + 7 FOR J J PICK R 7 PICK * + S 8 PICK * + 7 ROLL DROP DUP 6 ROLLD R 3 PICK * + S 4 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