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| Neopolys | |
| v8.1 | |
| (c)APO | |
| IMPORTANT !!! | |
| THIS VERSION WORKS ONLY IN THE HP49G AND HAS BEEN TESTED | |
| ONLY IN THE ROM VERSIONS 1.10 AND 1.14-2 | |
| ( The definitive polynomial lib for de HP48-G, GX, and ) | |
| ( now for the HP49G too ) | |
| ( WITH LAPLACE AND Z TRANSFORMS ) | |
| Fist of all: | |
| English is not my native language so I'd like to apologize for | |
| the errors in this doc. | |
| I'd like to thank all people who spend a moment of their life | |
| sending me one or another comment about this lib or any other one. | |
| THANKS !!! | |
| THE FILES | |
| ========= | |
| NeopV8x1esp.txt Spanish manual. | |
| NeopV8x1eng.txt Just what you're reading now. | |
| NeopV8x1.lib Library with Neopolys (v8.1) | |
| WARNING !! | |
| SOME COMMANDS HAVE IMPORTANT NOTES, PLEASE READ THEM. | |
| INTRODUCTION | |
| ============ | |
| 1. Polynomials can be represented: | |
| a) by their coefs in descendant order of exponent, which can | |
| be: | |
| reals : { 2 0 4 -500 } = '2·x^3+x-500' | |
| complexes : { (1,0) (3,-6) 2 } | |
| symbolics : { '1/3' 6 '2*(A,B)' '4/(B+C)' A } | |
| b) by their roots which can also be: | |
| reals : { { 1 2 3 4 } } | |
| complexes : { { (1,2) (3,4) 5 (6,0) } } | |
| symbolics : { { A B (0,-2) 'COS(E)' '2*INV(F)' 4 } } | |
| c) in any other case, the element at the corresponding level | |
| is converted to a polynomial (single list with one coefficient). | |
| For example: | |
| 2: 2 P+ | |
| 1: 'X' ->->->->->->->-> 1: { 'X+2' } | |
| NOTE: | |
| If in the working directory or in upper directories | |
| any variables with the same name that any variables which | |
| may appear in the expressions of the polynomial coefs are | |
| found, and if the content of that variable is a number or | |
| an expression with numerical result, its value is conside- | |
| red in the calculations instead of the symbolic variable | |
| name. | |
| Personally I've found this utility very useful because | |
| it allows some interesting studies of the polynomials accor- | |
| ding to the variation of their coefficients. | |
| 2. Polynomial fractions are represented with their numerator at | |
| level 2 and their denominator at level 1. | |
| 3. All commands work with numeric or symbolic coefficients except | |
| 'Roots' which only works with numeric values or symbolic expres- | |
| sions (as for example '3/4') that give numerical results, except | |
| when working with polynomials with degree less or equal than 2 | |
| in which case it can return the symbolic roots of that polynomial. | |
| 4. Polynomial '0' is represented by a null list ( { } ), but you can | |
| also enter another type of expressions like | |
| { 0 } or { (0,0) 0 0 '2*A-A-A' } | |
| (choose the one you prefer). | |
| 5. The polynomial { { } } is considered as the null polynomial { }. | |
| 6. The result of the commands depends on the state of the next user | |
| flags | |
| flag When active | |
| ---- ------------------------------------ | |
| 2 ........... Symbolic results are automatically simplified. | |
| Simplifies polynomial fractions when posible. | |
| Operations are slower mainly when working with | |
| symbolics. | |
| Some complicated expressions may not be entirely | |
| simplified. | |
| 3 ........... After each operation the command XQ. | |
| 4 ........... This flag is used only by the '->Z' command. | |
| When activated, the output of that command is | |
| directly expressed as a polynomial fraction. When | |
| not, it's expressed in an alternative form (See | |
| '->Z' below). | |
| You can activate or deactivate any user flag either typing | |
| N SF -> set flag number N | |
| N CF -> clear flag number N | |
| Nevertheless the Neopolys command 'flgs' make easy the activation/ | |
| deactivation of user flags 2 and 3. Just pres 'flgs' and see. | |
| THE COMMANDS | |
| ============ | |
| I think it's very obvious what each command does. Nevertheless | |
| I've decided to explain each of them. | |
| * P+ : Adds polynomials: | |
| 2: { 0 1 '3/4' } P+ | |
| 1: '-3/4' ->->->->->->->-> 1: { 1 0 } | |
| 2: { { '3/A' (1,1) 0 } } P+,flg3,flg2 1: { 1 '(0,1)-3/A' '2 | |
| 1: { 1 '2/3' (4,0) } ->->->->->->->-> /3+(3,3)/A' 4 } | |
| * P- : Subtracts polynomials: | |
| 2: { 1 2 3 4 } P-,flg2 1: { '2+A+B+C' '3+(-B- | |
| 1: { { A B C } } ->->->->->->->-> C)*A-B*C''4+A*B*C'} | |
| * P* : Multiplies polynomials: | |
| 2: { { 1 2 } } P* 1: { 1 -9 31 -51 40 | |
| 1: { 1 -6 11 -6 } ->->->->->->->-> -12 } | |
| * P/ : Divides polynomials: | |
| 2: { { A B 'C/3' } } P/,flg2 2: { }(remainder) | |
| 1: { { B 'C/3 } } ->->->->->->->-> 1: { 1 '-A'}(quotient) | |
| * P^ : Polynomial to integer power (or to expression with integer | |
| result): | |
| 2: { 1 '-A' } P^,flg2 1: { 1 '-(2*A)' 'A^2' | |
| 1: '(5*C-C)/(2*C)' ->->->->->->->-> } | |
| 2: { { 1 2 3 } } P^ 2: { 1 } | |
| 1: -2 ->->->->->->->-> 1: { 1 -12 58 -144 193 | |
| -132 36 } | |
| NOTES: | |
| Sometimes symbolic powers may not be completely simplified | |
| and cause an error. | |
| If you give a list at level 1 the first element of that list | |
| is considered as the power. | |
| * Pval : Evaluates polynomials using the Horner's method: | |
| 2: { 5 4 3 2 1 } Pval | |
| 1: (0,-1) ->->->->->->->-> 1: (3,2) | |
| 2: { { A B 1 } } Pval 1: '(1+C/3-1)*(1+C/3-B | |
| 1: '1+C/3' ->->->->->->->-> )*(1+C/3-A)' | |
| Next commands (which work with polynomical fractions) consider | |
| posible divides by zero. Remember that polynomial fractions requires | |
| two levels, one for numerator and another one for denominator. | |
| NOTE: | |
| All commands that return polynomial fractions always return | |
| two lists (although denominator is 1), EXCEPT IF THE RESULT IS ZERO, | |
| IN WHICH CASE THEY RETURN JUST ONE NULL LIST AT LEVEL 1. | |
| * F+ : Adds polynomial fractions. | |
| 4: { 1 2 } | |
| 3: { { 3 4 5 } } F+ | |
| 2: { 6 7 A } ->->->->->->->-> 2: { 6 8 'A+2' } | |
| 1: { 1 -12 47 -60 } 1: { 1 -12 47 -60 } | |
| * F- : Subtracts polynomial fractions. | |
| 4: 1 | |
| 3: { 2 } F-,flg2,flg3 | |
| 2: { { 3 } } ->->->->->->->-> 2: { '-1/4' '5/4' } | |
| 1: 4 1: { 1 } | |
| If you have not understood the concepts of working with fractions | |
| think that this example is the same that: | |
| 1 (s-3) (1/4)·s+(5/4) | |
| --- - ------- = --------------- = (1/4)·s+(5/4) | |
| 2 4 1 | |
| * F* : Multiplies polynomial fractions. | |
| * F/ : Divides polynomial fractions. | |
| * F^ : Polynomial fractions to integer power. It allows the same kind | |
| of symbolic expressions that P^ (see P^ above). | |
| * Fval : Evaluates polynomial fractions. | |
| * A->P : Converts an algebraic expression to its polynomial represen- | |
| tation. | |
| 2: 'A*s^2+B/(3*s)' A->P 2: { 'A*3' 0 0 B } | |
| 1: 's' ->->->->->->->-> 1: { 3 0 } | |
| NOTE: | |
| Keep in mind that the symbolic expression can be fractional or | |
| not, making the result be also fractional (two lists) or not (just | |
| one list). | |
| * P->A : Symbolic expression -> algebraic: | |
| 2: { A '1/3' (1,1) } P->A,flg3 | |
| 1: 'z' ->->->->->->->-> 1: 'A*z^2+1/3*z+(1,1)' | |
| 2: { { 1 2 3 } } P->A 1: '(S+3)^3-6*(S+3)^2+ | |
| 1: 'S+3' ->->->->->->->-> 11*(s+3)-6' | |
| NOTE: | |
| Giving a real or a complex value at level 1 you'll get the | |
| value of the polynomial evaluated at that point. | |
| * F->A : Polynomial fraction -> algebraic expression. | |
| 3: { { A B } } F->A | |
| 2: { 1 2 3 } ->->->->->->->-> 1: '(X^2+(-B-A)*X-B*-A | |
| 1: 'X' )/(X^2+2*X+3)' | |
| 3: { { 1 1 3 } } F->A | |
| 2: { { 1 2 } } ->->->->->->->-> 1: '(X^2-4*X+3)/(X-2)' | |
| 1: 'X' | |
| * Pd : Derivative of a polynomial: | |
| Pd,flg2 | |
| 1: { { 1 1 '2/(A+B)' } ->->->->->->->-> 1: { 3 '2*(-2-2/(A+B)) | |
| } ' '1+4/(A+B)' } | |
| * PS : Returns the primitive of the polynomial at level 1. | |
| * Fd : Derivative of a polynomial fraction. | |
| * Pcoef : Returns the coefficients of the polynomial whose roots are | |
| given at level 1. | |
| 1: { { 'A+B' 2 '1/3' } Pcoef,flg3 1: { 1 '-7/3+(-A-B)' ' | |
| } ->->->->->->->-> 2/3-7/3*(-A-B)' '2/ | |
| 3*(-A-B)' } | |
| Pcoef,flg3 | |
| 1: '2/3' ->->->->->->->-> 1: { 1 '-(2/3)' } | |
| * Fsimp : Simplifies polynomial fractions. | |
| 2: { { 1 2 } } Fsimp 2: { 1 } | |
| 1: { 1 -6 11 -6 } ->->->->->->->-> 1: { 1 -3 } | |
| 2: { { A B } } Fsimp 2: { 1 } | |
| 1: { { A B } } ->->->->->->->-> 1: { 1 } | |
| NOTE: | |
| Simplification of polynomials with a lot of symbolic coefs | |
| can spend a lot of time or hang. | |
| * GCD : Calculates the greatest common denominator of 2 polynomials. | |
| 2: { { 0 (1,0) A } } GCD,flg2 | |
| 1: { { 1 A } } ->->->->->->->-> 1: { 1 '-A-1' A } | |
| 2: { 1 2 3 4 5 } GCD | |
| 1: { 1 2 3 4 } ->->->->->->->-> 1: { 1 } | |
| (there is no GCD) | |
| * Roots : Returns the numeric roots of the polynomial at level 1, or | |
| the symbolic one for polynomials with degree less or equal than 2. | |
| Roots | |
| 1: { 1 -15 85 -225 274 ->->->->->->->-> 1: { { 1 2 3 4 5 } } | |
| -120 } | |
| 1: { 1 '-(7/3)' '14/9' Roots,flg3 1: { { '1/3' '2/3' '4/ | |
| '-(8/27)' } ->->->->->->->-> 3' } } | |
| NOTE: | |
| I remember you that you can obtain SYMBOLIC roots for polyno- | |
| mials whose degree is less or equal than 2 (lists with less than | |
| 4 elements). | |
| You can also calaulate the polynomial roots for a polynomial | |
| with symbolic coefs which gives numerical results. For example, | |
| if you have stored the values 1, -3 and -12 y a variables at the | |
| current directory or any directory upon it, with the names 'A', | |
| 'B' and 'C' respectively, you could do this: | |
| Roots | |
| 1: { A '-A+3*B' '-A+B*C' ->->->->->->->-> 1: { { 1 2 3 4 } } | |
| '3*C+4*B-2*A' '-2*C' } | |
| This also would work if 'A', 'B' y 'C' were temporary varia- | |
| bles in a program. For example, the next user-RPL program | |
| << 1 -3 -12 -> A B C | |
| << | |
| 'A' | |
| '-A+3*B' | |
| '-A+B*C' | |
| '3*C+4*B-2*A' | |
| '-2*C' | |
| 5 ->LIST | |
| Roots | |
| >> | |
| >> | |
| would also return the same that in the previous example: | |
| 1: { { 1 2 3 4 } } | |
| * pf : Partial fraction decomposition. | |
| The list at level 2 represents the coefficients that corres- | |
| ponding the given root at the same position in the list at | |
| level 1. | |
| 2: { 1 } pf 2: { '1/(A-B)' '1/(... | |
| 1: { { A B } } ->->->->->->->-> 1: { { A B } } | |
| For multiple roots , the coefs given in the list at level 2 | |
| are sorted from lower to higher exponent of the corresponding | |
| root, that is: | |
| 2: { { 1 2 } } pf 2: { 1 0 -1 -1 0 } | |
| 1: { { 1 2 1 2 1 } } ->->->->->->->-> 1: { { 2 2 1 1 1 } } | |
| meaning: | |
| (x-1)·(x-2) | |
| ----------------------------- = | |
| (x-1)·(x-2)·(x-1)·(x-2)·(x-1) | |
| 1 0 -1 -1 0 | |
| = ------- + --------- + ------- + --------- + --------- | |
| (x-2) (x-2)^2 (x-1) (x-1)^2 (x-1)^3 | |
| * flgs : Activates/deactivates flags 2 and 3 alternatively. For flag | |
| 4 use the form explained at the beginning of this doc, that | |
| is, 4 CF (clears user flag 4) and 4 SF (sets user flag 4). | |
| * DoLis : Applies the program at level 1 to the list at level 2: | |
| 2: { 1 2 3 4 5 } DoLis | |
| 1: << 1 + >> ->->->->->->->-> 1: { 2 3 4 5 6 } | |
| NOTE: | |
| This command now also works with symbolic matrices (list of lists) | |
| and polynomials given by their roots. | |
| 2: { { '1/3' } { '2/4 } } DoLis | |
| 1: << ->NUM >> ->->->->->->->-> 1: { { 0.333333333333 } | |
| { .5 } } | |
| 2: { { 2 6 8 } } DoLis | |
| 1: << 2 / >> ->->->->->->->-> 1: { { 1 3 4 } } | |
| * RevLis : Fast! reverse of the elements in a list. | |
| RevLis | |
| 1: { 1 2 3 4 5 6 } ->->->->->->->-> 1: { 6 5 4 3 2 1 } | |
| NOTE: | |
| As the command above, DoLis, this one also works with symbolic | |
| matrices and polynomials given by their roots. | |
| RevLis | |
| 1: { { 1 2 } { 3 4 } } ->->->->->->->-> 1: { { 4 3 } { 2 1 } } | |
| RevLis | |
| 1: { { 1 2 3 4 } } ->->->->->->->-> 1: { { 4 3 2 1 } } | |
| * Q-> : This command is the inverse of '->Q'. It converts expressions | |
| that have numerical and fractional terms to numbers. It is useful | |
| when you've activated flag 3 to obtain fractional results and you | |
| don't like the result because it contains fractions like '1234/5678' | |
| or when you want to convert symbolic (with numerical result) ex- | |
| pressions to numbers. Let's see: | |
| Q-> | |
| 1: { '1/3' 'A+2/5*B' } ->->->->->->->-> 1: { .333333333333 'A+ | |
| .4*B' } | |
| Let be 'V' the square root and 'pi' the symbol that represents | |
| the number pi (3.1415...), then if for example you executed Roots | |
| and obtained the roots { { 'V3''pi' } }, you can apply Q-> to get | |
| the numerical values | |
| Q-> | |
| 1: { { 'pi' 'V3' } } ->->->->->->->-> 1: { { 3.14159265359 | |
| 1.73205080757 } } | |
| It's also considered the posibility of the variables which | |
| intervene in the expressions of the polynomial coefs to be in the | |
| current directory or upper directories and that they contain num- | |
| eric expressions. For example, try to create two variables in the | |
| current directory, one called 'A' which contains the number 2 and | |
| one called 'B' which contains '6/2' ( simply by entering | |
| 2 'A' STO | |
| '6/2' 'B' STO | |
| ). | |
| And now, with those variables created think about the result of | |
| the next example: | |
| Q-> | |
| 1: { A 'A+B' 'A*B+C' } ->->->->->->->-> 1: { 2 5 '6+C' } | |
| (A=2,B='6/2') | |
| -o-o- | |
| The next commands are only available when you install the file | |
| .trn containing the commands to manage Laplace and Z transform. | |
| * ->L : Returns the Laplace transform of expressions formed by combina- | |
| tions of | |
| · K (constant) | |
| · t^n | |
| · e^( a·t ± b ) | |
| · sin( a·t ± b ) | |
| · cos( a·t ± b ) | |
| · sinh( a·t ± b ) | |
| · cosh( a·t ± b ) | |
| NOTE: | |
| Remember that expressions must be given in terms of 't'. | |
| 1: 'e^-t*COS(3*t+2)+2* ->L,DEG 2: { .999390827019 6.... | |
| e^-(2*t)*t' ->->->->->->->-> 1: { 1 6 22 48 40 } | |
| ->L,flg2 2: { 'COS(b)''-(2*SI... | |
| 1: 't*COS(a*t+b)' ->->->->->->->-> 1: { 1 0 '2*a^2' 0 'a^ | |
| 4' } | |
| * L-> : Returns the inverse Laplace transform of the polynomial frac- | |
| tion given by the lists at levels 1 and 2 ( with numerator | |
| greater than numerator ). | |
| 2: { 1 } L-> 1: '-(.5*e^-t*COS(t))+ | |
| 1: { { (-1,1) (-1,-1) ->->->->->->->-> 0.5*e^-(2*t)+.5*t*e^ | |
| -2 -2 } } -(2*t)' | |
| NOTE: | |
| inverse Laplace transforms of hyperbolic functions are returned | |
| with exponentials. Remember that | |
| e^t - e^(-t) | |
| sinh(t) = -------------- | |
| 2 | |
| e^t + e^(-t) | |
| cosh(t) = -------------- | |
| 2 | |
| * ->Z : Due to the extensive use of Z transform calculations as from | |
| functions in "s" (that is, as from Laplace transforms), this | |
| command expects a polynomial fraction at levels 1 and 2 that | |
| is the result of a Laplace transform. | |
| Since Z transforms normally include exponentials which | |
| depends on the sampling period , results can be composed by | |
| very large and confused expressions . This is the main reason | |
| that made me think about expressing the result in an alterna- | |
| tive form. Thus depending on the state of user flag 4 you can | |
| choose between obtain the fractional result or the alternative | |
| one (if that flag is set or not respectively). | |
| I don't know if you'll find the alternative form useful, it | |
| really does for me. | |
| Let's see it with an example | |
| EXAMPLE | |
| ------- | |
| If x(t) is | |
| x(t)=1-t·e^(-t) | |
| what will be its Z transform ? | |
| -0-0- | |
| SOL: | |
| --- | |
| X(z) = Z[x(t)] = Z[x(kT)] = Z[1-kT·e^(-kT)] | |
| To calculate this Z transform you enter '1-t·e^(-t)' at le- | |
| vel 1 and do '->L' obtaining the corresponding Laplace trans- | |
| form | |
| 2: { 1 1 1 } | |
| 1: { 1 2 1 0 } | |
| which as explained in the '->L' command means | |
| s^2 + s + 1 | |
| X(s) = L[x(t)] = ----------------- | |
| s^3 + 2·s^2 + s | |
| and at this point you can do either: | |
| a) set flag 4 and press '->Z', obtaining the whole | |
| expression for the Z transform of X(s): | |
| 2: { 1 '-(T*e^-T)-2*e... | |
| 1: { 1 '-1-2*e^-T' 'e^ | |
| (-T-T)+2*e^-T' '-e^ | |
| (-T-T}' } | |
| meanning: | |
| z^3 + [-T·e^(-T)-2·e^(-T)]·z^2 + [T·e^(-T)+e^(-2T)]·z | |
| X(z) = ----------------------------------------------------------- | |
| z^3 + [-1-2·e^(-T)]·z^2 + (e^(-2T)+2·e^(-T)]·z + (-e^(-2T)) | |
| Some people likes negative powers of z ... simply divide | |
| both numerator and denominator by z^3 ... | |
| Or | |
| b) clear flag 4 and press '->Z', obtaining an alternative | |
| form for the result: | |
| 2: { { 1 0 } { '-(T*e... | |
| 1: { { 0 1 } { '-T' 2 | |
| } } | |
| meanning | |
| z -T·e^(-T)·z | |
| X(z)= ------------- + --------------- (*) | |
| (z - e^0)^1 (z - e^(-T))^2 | |
| A A A A | |
| That is, the lists contained in the list at level 2 are | |
| polynomials in descendent order of z and represents the | |
| numerators of the partial fraction decomposition. The | |
| lists contained in the one at level 1 are in the form | |
| { X Y } | |
| where X is the exponent which has to be applied to "e" | |
| and then substracted to "z" to obtain an expression like | |
| ( z - e^X ) | |
| and Y is the power needed to obtain the whole expression | |
| of the denominator, something like: | |
| ( z - e^X )^Y | |
| ( In (*) the A's mark X and Y for each term ) | |
| Do you understand? I think it's not too complicated and | |
| after a bit of practicing with it you'll notice how much | |
| useful this form is (at least it does for me). | |
| Would you like to particularize the Z transform for a value | |
| of the sampling period "T" ? | |
| NO PROBLEM !!! | |
| This command does it for you. Give it X(s)(like always in a poly- | |
| nomial fraction but now at levels 2 and 3) and the real value of | |
| the sampling period "T" at level 1, and it'll return back to you | |
| the Z transform particularized for that period. | |
| For example: | |
| Once you've got X(s) in the form: | |
| 2: { 1 1 1 } | |
| 1: { 1 2 1 0 } | |
| enter T at level 1. | |
| E.g. ( for T = 0.5 seconds ) | |
| 3: { 1 1 1 } | |
| 2: { 1 2 1 0 } | |
| 1: .5 | |
| And do '->Z' to obtain the Z transform. | |
| Did you obtain the alternative form and you'd like to get the | |
| whole one? | |
| NO PROBLEM !!! | |
| That's just what next command does: | |
| * ->ND: Given a Z transform in the alternative form returns the whole | |
| one. | |
| Continuing with the previous example... | |
| Once you've you the alternative expression | |
| 2: { { 1 0 } { '-(T*e... | |
| 1: { { 0 1 } { '-T' 2 | |
| } } | |
| press '->ND' and you'll obtain the polynomial fractions at | |
| levels 1 and 2 which represents the Z transform in its whole | |
| form. | |
| * Z-> : Returns the inverse Z transform of the polynomial fraction | |
| (expressed in descendent (and positive) powers of z) given at | |
| levels 1 and 2. | |
| EXAMPLE | |
| ------- | |
| To calculate the inverse Z transform of | |
| z z^-1 | |
| X(z) = --------------- = ------------------- | |
| z^2 + 3·z + 2 1 + 3·z^-1 + z^-2 | |
| enter | |
| 2: { 1 0 } | |
| 1: { 1 3 2 } | |
| and do 'Z->'. You'll get something like this: | |
| 1: (-1)^k-(-2)^k | |
| that means that | |
| x(k) = Z^-1[X(z)] = (-1)^k-(-2)^k | |
| NOTES (for '->Z' and 'Z->' ): | |
| * These two commands only works with polynomials WHOSE ROOTS HAVE | |
| A MULTIPLICITY LESS OR EQUAL THAN 4 ( which is enough in most ca- | |
| ses ). | |
| NOTICE THAT IT DOESN'T MEAN THAT YOU CAN'T WORK WITH | |
| POLYNOMIALS THAT ONLY HAVE FOUR COEFFICIENTS | |
| * You can also play with user flag 2 to obtain more simplified | |
| expressions. | |
| * Don't forget that you can keep on using the current notation of | |
| Neopolys for the polynomials passed to these commands. | |
| For example: in the 'Z->''s example, you could have given | |
| 2: { { 0 } } | |
| 1: { { -1 -2 } } | |
| and you'd have obtained the same result. | |
| * To particularize Z transforms for a given T you could also use | |
| the calculator 'MATCH's commands in combination with 'Dolis' ... | |
| I'll let you think about it... | |
| QUICK REFERENCE | |
| =============== | |
| The next one is a summary of each of the commands implemented in | |
| this version and all of their arguments and returns. If you have | |
| never seen this kind of notation it's very simple to understand: the | |
| terms to the left of the simbol "->->" represents the arguments | |
| passed to the command and their order in the stack, and the one to | |
| the right the results. For example: | |
| P+ : (lrts,any) (lrts,any) ->-> list | |
| ( Stack level -> (2) (1) ->-> (1) ) | |
| means that the P+ command takes either a polynomial or any object at | |
| level 2 ( position (2) ) and the same kind of arguments for level 1, | |
| returning a polynomial at level 1. | |
| The meaning of each simbol is: | |
| list = A list with the coefficientes of a polynomial. | |
| E.g.: { 1 2 3 4 } | |
| roots = A double list with the roots of the polynomial. | |
| E.g.: { { 1 2 3 4 } } | |
| lrts = Any of the last two one. | |
| E.g.: { 1 2 3 } or { { 1 2 3 } } | |
| listlst = A list of lists (alternative form of '->Z') | |
| E.g.: { { 1 0 } { 1 3 } } | |
| simb = Simbolic expression. | |
| E.g.: 'x^2+1/A' | |
| simbn = Simbolic expression with numerical result: | |
| E.g.: '1/3+2^3' ( = 8.3333... ) | |
| var = Variable name. | |
| E.g.: 's' | |
| num = Real or complex number. | |
| E.g.: 1) 123 | |
| 2) (1,-23.12) | |
| proc = Procedure. | |
| E.g.: << 2 * >> | |
| any = Any object. | |
| And the reference for each command is: | |
| P+ : ADD POLYNOMIALS | |
| (lrts,any) (lrts,any) ->-> list | |
| P- : SUBSTRACT POLYNOMIALS | |
| (lrts,any) (lrts,any) ->-> list | |
| P* : MULTIPLIES POLYNOMIALS | |
| (lrts,any) (lrts,any) ->-> list | |
| P/ : DIVIDES POLYNOMIALS | |
| (lrts,any) (lrts,any) ->-> list(resto) list(cociente) | |
| P^ : POLYNOMIAL TO INTEGER POWER | |
| (lrts,any) (num,simbn,list) ->-> list | |
| Pval : EVALUATES POLYNOMIALS | |
| (lrts,any) (num,simbn,simb,var) ->-> num | |
| F+ : (lrts,any) (lrts,any) (lrts,any) (lrts,any) ->-> list list | |
| ( ó { } ) | |
| F- : (lrts,any) (lrts,any) (lrts,any) (lrts,any) ->-> list list | |
| ( ó { } ) | |
| F* : (lrts,any) (lrts,any) (lrts,any) (lrts,any) ->-> list list | |
| ( ó { } ) | |
| F/ : (lrts,any) (lrts,any) (lrts,any) (lrts,any) ->-> list list | |
| ( ó { } ) | |
| F^ : (lrts,any) (lrts,any) (num,simbn) ->-> list list | |
| ( ó { } ) | |
| Fval : (lrts,any) (lrts,any) (num,simbn,simb,var) ->-> num | |
| A->P : ALGEBRAIC EXPRESSION TO POLYNOMIAL | |
| (simb,num,var) var ->-> list | |
| list list | |
| P->A : POLYNOMIAL TO ALGEBRAIC EXPRESSION | |
| (lrts,any) (var,num,simb) ->-> simb | |
| F->A : POLYNOMIAL FRACTION TO ALGEBRAIC EXPRESSION | |
| (lrts,any) (lrts,any) (var,num,simb) ->-> simb | |
| Pd : DERIVES POLYNOMIALS | |
| (lrts,any) ->-> list | |
| PS : INTEGRATES POLYNOMIALS | |
| (lrts,any) ->-> list | |
| Fd : DERIVES POLYNOMIAL FRACTIONS | |
| (lrts,any) (lrts,any) ->-> list list | |
| { } | |
| Pcoef : POLYNOMIAL COEFFICIENTS GIVEN THE ROOTS | |
| (roots,any) ->-> list | |
| Fsimp : POLYNOMIAL FRACTION SIMPLIFICATION | |
| (lrts,any) (lrts,any) ->-> list list | |
| { } | |
| GCD : GREATEST COMMON DENOMINATOR | |
| (lrts,any) (lrts,any) ->-> list | |
| Roots : POLYNOMIAL ROOTS GIVEN THE COEFFICIENTS | |
| (lrts,any) ->-> roots | |
| pf : PARTIAL FRACTION DECOMPOSITION | |
| (lrts,any) (lrts,any) -> -> list roots | |
| flgs : USER FLAGS 2 AND 3 STATE CHANGING | |
| ->-> (modified state of user flags 2 and 3) | |
| DoLis : APPLIES A PROCEDURE TO EACH POLYNOMIAL COEF | |
| (lrts,listlst) proc ->-> (lrts,listlst) | |
| RevLis : REVERSE LISTS | |
| (lrts,listlst) ->-> (lrts,listlst) | |
| Q-> : CONVERTS NUMERICAL FRACTIONS AND NUMERICAL EXPRESSIONS TO NUMBERS | |
| REPLACING THE CONTENTS OF NUMERIC VARIABLES LOCATED IN THE CURRENT | |
| OR UPPER DIRECTORIES | |
| (lrts,listlst) ->-> (lrts,listlst) | |
| ->L : LAPLACE TRANSFORM | |
| symb(in terms of t) ->-> list list | |
| { } | |
| L-> : INVERSE LAPLACE TRANSFORM | |
| (lrts,any) (lrts,any) ->-> symb | |
| ->Z : Z TRANSFORM | |
| lrts lrts ->-> listlst listlst (flag 4 not set) | |
| ->-> list list (flag 4 set) | |
| lrts lrts num(real) ->-> listlst listlst (flag 4 not set) | |
| ->-> list list (flag 4 set) | |
| Z-> : INVERSE Z TRANSFORM | |
| (lrts,any) (lrts,any) ->-> symb | |
| ->ND : ALTERNATIVE EXPRESSION TO WHOLE EXPRESSION | |
| listlst listlst ->-> list list | |
| THE AUTOR | |
| ========= | |
| I would be very glad if you send me any comment about the library | |
| or about possible bugs that you can find when using this lib. You can | |
| find me at | |
| [email protected] | |
| FINAL NOTE | |
| ========== | |
| There is a library called Bode-Routh which allows you to make Bo- | |
| de plots, Nyquist, root locus, etc... That library uses Neopolys to | |
| work and is available at the same locations that Neopolys so you can | |
| find it easily if you are reading this document now. |
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