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This is maxima.info, produced by makeinfo version 4.1 from maxima.texi. This is a Texinfo Maxima Manual Copyright 1994,2001 William F. Schelter START-INFO-DIR-ENTRY * Maxima: (maxima). A computer algebra system. END-INFO-DIR-ENTRY File: maxima.info, Node: Definitions for Integration, Prev: Introduction to Integration, Up: Integration Definitions for Integration =========================== - Function: CHANGEVAR (exp,f(x,y),y,x) makes the change of variable given by f(x,y) = 0 in all integrals occurring in exp with integration with respect to x; y is the new variable. (C1) 'INTEGRATE(%E**SQRT(A*Y),Y,0,4); 4 / [ SQRT(A) SQRT(Y) (D1) I (%E ) dY ] / 0 (C2) CHANGEVAR(D1,Y-Z^2/A,Z,Y); 2 SQRT(A) / [ Z 2 I Z %E dZ ] / 0 (D4) --------------------- A CHANGEVAR may also be used to changes in the indices of a sum or product. However, it must be realized that when a change is made in a sum or product, this change must be a shift, i.e. I=J+ ..., not a higher degree function. E.g. (C3) SUM(A[I]*X^(I-2),I,0,INF); INF ==== \ I - 2 (D3) > A X / I ==== I = 0 (C4) CHANGEVAR(%,I-2-N,N,I); INF ==== \ N (D4) > A X / N + 2 ==== N = - 2 - Function: DBLINT ('F,'R,'S,a,b) a double-integral routine which was written in top-level macsyma and then translated and compiled to machine code. Use LOAD(DBLINT); to access this package. It uses the Simpson's Rule method in both the x and y directions to calculate /B /S(X) | | | | F(X,Y) DY DX . | | /A /R(X) The function F(X,Y) must be a translated or compiled function of two variables, and R(X) and S(X) must each be a translated or compiled function of one variable, while a and b must be floating point numbers. The routine has two global variables which determine the number of divisions of the x and y intervals: DBLINT_X and DBLINT_Y, both of which are initially 10, and can be changed independently to other integer values (there are 2*DBLINT_X+1 points computed in the x direction, and 2*DBLINT_Y+1 in the y direction). The routine subdivides the X axis and then for each value of X it first computes R(X) and S(X); then the Y axis between R(X) and S(X) is subdivided and the integral along the Y axis is performed using Simpson's Rule; then the integral along the X axis is done using Simpson's Rule with the function values being the Y-integrals. This procedure may be numerically unstable for a great variety of reasons, but is reasonably fast: avoid using it on highly oscillatory functions and functions with singularities (poles or branch points in the region). The Y integrals depend on how far apart R(X) and S(X) are, so if the distance S(X)-R(X) varies rapidly with X, there may be substantial errors arising from truncation with different step-sizes in the various Y integrals. One can increase DBLINT_X and DBLINT_Y in an effort to improve the coverage of the region, at the expense of computation time. The function values are not saved, so if the function is very time-consuming, you will have to wait for re-computation if you change anything (sorry). It is required that the functions F, R, and S be either translated or compiled prior to calling DBLINT. This will result in orders of magnitude speed improvement over interpreted code in many cases! The file SHARE1;DBLINT DEMO can be run in batch or demo mode to illustrate the usage on a sample problem; the file SHARE1;DBLNT DEMO1 is an extension of the DEMO which also makes use of other numerical aids, FLOATDEFUNK and QUANC8. Please send all bug notes and questions to LPH - Function: DEFINT (exp, var, low, high) DEFinite INTegration, the same as INTEGRATE(exp,var,low,high). This uses symbolic methods, if you wish to use a numerical method try ROMBERG(exp,var,low,high). - Function: ERF (X) the error function, whose derivative is: 2*EXP(-X^2)/SQRT(%PI). - Variable: ERFFLAG default: [TRUE] if FALSE prevents RISCH from introducing the ERF function in the answer if there were none in the integrand to begin with. - Variable: ERRINTSCE default: [TRUE] - If a call to the INTSCE routine is not of the form EXP(A*X+B)*COS(C*X)^N*SIN(C*X) then the regular integration program will be invoked if the switch ERRINTSCE[TRUE] is TRUE. If it is FALSE then INTSCE will err out. - Function: ILT (exp, lvar, ovar) takes the inverse Laplace transform of exp with respect to lvar and parameter ovar. exp must be a ratio of polynomials whose denominator has only linear and quadratic factors. By using the functions LAPLACE and ILT together with the SOLVE or LINSOLVE functions the user can solve a single differential or convolution integral equation or a set of them. (C1) 'INTEGRATE(SINH(A*X)*F(T-X),X,0,T)+B*F(T)=T**2; T / [ 2 (D1) I (SINH(A X) F(T - X)) dX + B F(T) = T ] / 0 (C2) LAPLACE(%,T,S); A LAPLACE(F(T), T, S) (D2) --------------------- 2 2 S - A 2 + B LAPLACE(F(T), T, S) = -- 3 S (C3) LINSOLVE([%],['LAPLACE(F(T),T,S)]); SOLUTION 2 2 2 S - 2 A (E3) LAPLACE(F(T), T, S) = -------------------- 5 2 3 B S + (A - A B) S (D3) [E3] (C4) ILT(E3,S,T); IS A B (A B - 1) POSITIVE, NEGATIVE, OR ZERO? POS; 2 SQRT(A) SQRT(A B - B) T 2 COSH(------------------------) B (D4) F(T) = - -------------------------------- A 2 A T 2 + ------- + ------------------ A B - 1 3 2 2 A B - 2 A B + A - Function: INTEGRATE (exp, var) integrates exp with respect to var or returns an integral expression (the noun form) if it cannot perform the integration (see note 1 below). Roughly speaking three stages are used: * (1) INTEGRATE sees if the integrand is of the form F(G(X))*DIFF(G(X),X) by testing whether the derivative of some subexpression (i.e. G(X) in the above case) divides the integrand. If so it looks up F in a table of integrals and substitutes G(X) for X in the integral of F. This may make use of gradients in taking the derivative. (If an unknown function appears in the integrand it must be eliminated in this stage or else INTEGRATE will return the noun form of the integrand.) * (2) INTEGRATE tries to match the integrand to a form for which a specific method can be used, e.g. trigonometric substitutions. * (3) If the first two stages fail it uses the Risch algorithm. Functional relationships must be explicitly represented in order for INTEGRATE to work properly. INTEGRATE is not affected by DEPENDENCIES set up with the DEPENDS command. INTEGRATE(exp, var, low, high) finds the definite integral of exp with respect to var from low to high or returns the noun form if it cannot perform the integration. The limits should not contain var. Several methods are used, including direct substitution in the indefinite integral and contour integration. Improper integrals may use the names INF for positive infinity and MINF for negative infinity. If an integral "form" is desired for manipulation (for example, an integral which cannot be computed until some numbers are substituted for some parameters), the noun form 'INTEGRATE may be used and this will display with an integral sign. (See Note 1 below.) The function LDEFINT uses LIMIT to evaluate the integral at the lower and upper limits. Sometimes during integration the user may be asked what the sign of an expression is. Suitable responses are POS;, ZERO;, or NEG;. (C1) INTEGRATE(SIN(X)**3,X); 3 COS (X) (D1) ------- - COS(X) 3 (C2) INTEGRATE(X**A/(X+1)**(5/2),X,0,INF); IS A + 1 POSITIVE, NEGATIVE, OR ZERO? POS; IS 2 A - 3 POSITIVE, NEGATIVE, OR ZERO? NEG; 3 (D2) BETA(A + 1, - - A) 2 (C3) GRADEF(Q(X),SIN(X**2)); (D3) Q(X) (C4) DIFF(LOG(Q(R(X))),X); d 2 (-- R(X)) SIN(R (X)) dX (D4) -------------------- Q(R(X)) (C5) INTEGRATE(%,X); (D5) LOG(Q(R(X))) (Note 1) The fact that MACSYMA does not perform certain integrals does not always imply that the integral does not exist in closed form. In the example below the integration call returns the noun form but the integral can be found fairly easily. For example, one can compute the roots of `X^3+X+1 = 0' to rewrite the integrand in the form 1/((X-A)*(X-B)*(X-C)) where A, B and C are the roots. MACSYMA will integrate this equivalent form although the integral is quite complicated. (C6) INTEGRATE(1/(X^3+X+1),X); / [ 1 (D6) I ---------- dX ] 3 / X + X + 1 - Variable: INTEGRATION_CONSTANT_COUNTER - a counter which is updated each time a constant of integration (called by MACSYMA, e.g., "INTEGRATIONCONSTANT1") is introduced into an expression by indefinite integration of an equation. - Variable: INTEGRATE_USE_ROOTSOF default: [false] If not false then when the denominator of an rational function cannot be factored, we give the integral in a form which is a sum over the roots of the denominator: (C4) integrate(1/(1+x+x^5),x); / 2 [ x - 4 x + 5 I ------------ dx 2 x + 1 ] 3 2 2 5 ATAN(-------) / x - x + 1 LOG(x + x + 1) SQRT(3) (D4) ----------------- - --------------- + --------------- 7 14 7 SQRT(3) but now we set the flag to be true and the first part of the integral will undergo further simplification. (C5) INTEGRATE_USE_ROOTSOF:true; (D5) TRUE (C6) integrate(1/(1+x+x^5),x); ==== 2 \ (%R1 - 4 %R1 + 5) LOG(x - %R1) > ------------------------------- / 2 ==== 3 %R1 - 2 %R1 3 2 %R1 in ROOTSOF(x - x + 1) (D6) ---------------------------------------------------------- 7 2 x + 1 2 5 ATAN(-------) LOG(x + x + 1) SQRT(3) - --------------- + --------------- 14 7 SQRT(3) Note that it may be that we want to approximate the roots in the complex plane, and then provide the function factored, since we will then be able to group the roots and their complex conjugates, so as to give a better answer. - Function: INTSCE (expr,var) INTSCE LISP contains a routine, written by Richard Bogen, for integrating products of sines,cosines and exponentials of the form EXP(A*X+B)*COS(C*X)^N*SIN(C*X)^M The call is INTSCE(expr,var) expr may be any expression, but if it is not in the above form then the regular integration program will be invoked if the switch ERRINTSCE[TRUE] is TRUE. If it is FALSE then INTSCE will err out. - Function: LDEFINT (exp,var,ll,ul) yields the definite integral of exp by using LIMIT to evaluate the indefinite integral of exp with respect to var at the upper limit ul and at the lower limit ll. - Function: POTENTIAL (givengradient) The calculation makes use of the global variable POTENTIALZEROLOC[0] which must be NONLIST or of the form [indeterminatej=expressionj, indeterminatek=expressionk, ...] the former being equivalent to the nonlist expression for all right-hand sides in the latter. The indicated right-hand sides are used as the lower limit of integration. The success of the integrations may depend upon their values and order. POTENTIALZEROLOC is initially set to 0. - Function: QQ - The file SHARE1;QQ FASL (which may be loaded with LOAD("QQ");) contains a function QUANC8 which can take either 3 or 4 arguments. The 3 arg version computes the integral of the function specified as the first argument over the interval from lo to hi as in QUANC8('function name,lo,hi); . The function name should be quoted. The 4 arg version will compute the integral of the function or expression (first arg) with respect to the variable (second arg) over the interval from lo to hi as in QUANC8(<f(x) or expression in x>,x,lo,hi). The method used is the Newton-Cotes 8th order polynomial quadrature, and the routine is adaptive. It will thus spend time dividing the interval only when necessary to achieve the error conditions specified by the global variables QUANC8_RELERR (default value=1.0e-4) and QUANC8_ABSERR (default value=1.0e-8) which give the relative error test: |integral(function)-computed value|< quanc8_relerr*|integral(function)| and the absolute error test: |integral(function)-computed value|<quanc8_abserr. Do PRINTFILE(QQ,USAGE,SHARE1) for details. - Function: QUANC8 ('function name,lo,hi) An adaptive integrator, available in SHARE1;QQ FASL. DEMO and USAGE files are provided. The method is to use Newton-Cotes 8-panel quadrature rule, hence the function name QUANC8, available in 3 or 4 arg versions. Absolute and relative error checks are used. To use it do LOAD("QQ"); For more details do DESCRIBE(QQ); . - Function: RESIDUE (exp, var, val) computes the residue in the complex plane of the expression exp when the variable var assumes the value val. The residue is the coefficient of (var-val)**(-1) in the Laurent series for exp. (C1) RESIDUE(S/(S**2+A**2),S,A*%I); 1 (D1) - 2 (C2) RESIDUE(SIN(A*X)/X**4,X,0); 3 A (D2) - -- 6 - Function: RISCH (exp, var) integrates exp with respect to var using the transcendental case of the Risch algorithm. (The algebraic case of the Risch algorithm has not been implemented.) This currently handles the cases of nested exponentials and logarithms which the main part of INTEGRATE can't do. INTEGRATE will automatically apply RISCH if given these cases. ERFFLAG[TRUE] - if FALSE prevents RISCH from introducing the ERF function in the answer if there were none in the integrand to begin with. (C1) RISCH(X^2*ERF(X),X); 2 2 - X X 3 2 %E (%E SQRT(%PI) X ERF(X) + X + 1) (D1) ------------------------------------------ 3 SQRT(%PI) (C2) DIFF(%,X),RATSIMP; 2 (D2) X ERF(X) - Function: ROMBERG (exp,var,ll,ul) or ROMBERG(exp,ll,ul) - Romberg Integration. You need not load in any file to use ROMBERG, it is autoloading. There are two ways to use this function. The first is an inefficient way like the definite integral version of INTEGRATE: ROMBERG(<integrand>,<variable of integration>,<lower limit>, <upper limit>); Examples: ROMBERG(SIN(Y),Y,1,%PI); TIME= 39 MSEC. 1.5403023 F(X):=1/(X^5+X+1); ROMBERG(F(X),X,1.5,0); TIME= 162 MSEC. - 0.75293843 The second is an efficient way that is used as follows: ROMBERG(<function name>,<lower limit>,<upper limit>); Example: F(X):=(MODE_DECLARE([FUNCTION(F),X],FLOAT),1/(X^5+X+1)); TRANSLATE(F); ROMBERG(F,1.5,0); TIME= 13 MSEC. - 0.75293843 The first argument must be a TRANSLATEd or compiled function. (If it is compiled it must be declared to return a FLONUM.) If the first argument is not already TRANSLATEd, ROMBERG will not attempt to TRANSLATE it but will give an error. The accuracy of the integration is governed by the global variables ROMBERGTOL (default value 1.E-4) and ROMBERGIT (default value 11). ROMBERG will return a result if the relative difference in successive approximations is less than ROMBERGTOL. It will try halving the stepsize ROMBERGIT times before it gives up. The number of iterations and function evaluations which ROMBERG will do is governed by ROMBERGABS and ROMBERGMIN, do DESCRIBE(ROMBERGABS,ROMBERGMIN); for details. ROMBERG may be called recursively and thus can do double and triple integrals. Example: INTEGRATE(INTEGRATE(X*Y/(X+Y),Y,0,X/2),X,1,3); 13/3 (2 LOG(2/3) + 1) %,NUMER; 0.81930233 DEFINE_VARIABLE(X,0.0,FLOAT,"Global variable in function F")$ F(Y):=(MODE_DECLARE(Y,FLOAT), X*Y/(X+Y) )$ G(X):=ROMBERG('F,0,X/2)$ ROMBERG(G,1,3); 0.8193023 The advantage with this way is that the function F can be used for other purposes, like plotting. The disadvantage is that you have to think up a name for both the function F and its free variable X. Or, without the global: G1(X):=(MODE_DECLARE(X,FLOAT), ROMBERG(X*Y/(X+Y),Y,0,X/2))$ ROMBERG(G1,1,3); 0.8193023 The advantage here is shortness. Q(A,B):=ROMBERG(ROMBERG(X*Y/(X+Y),Y,0,X/2),X,A,B)$ Q(1,3); 0.8193023 It is even shorter this way, and the variables do not need to be declared because they are in the context of ROMBERG. Use of ROMBERG for multiple integrals can have great disadvantages, though. The amount of extra calculation needed because of the geometric information thrown away by expressing multiple integrals this way can be incredible. The user should be sure to understand and use the ROMBERGTOL and ROMBERGIT switches. - Variable: ROMBERGABS default: [0.0] (0.0B0) Assuming that successive estimates produced by ROMBERG are Y[0], Y[1], Y[2] etc., then ROMBERG will return after N iterations if (roughly speaking) (ABS(Y[N]-Y[N-1]) <= ROMBERGABS OR ABS(Y[N]-Y[N-1])/(IF Y[N]=0.0 THEN 1.0 ELSE Y[N]) <= ROMBERGTOL) is TRUE. (The condition on the number of iterations given by ROMBERGMIN must also be satisfied.) Thus if ROMBERGABS is 0.0 (the default) you just get the relative error test. The usefulness of the additional variable comes when you want to perform an integral, where the dominant contribution comes from a small region. Then you can do the integral over the small dominant region first, using the relative accuracy check, followed by the integral over the rest of the region using the absolute accuracy check. Example: Suppose you want to compute Integral(exp(-x),x,0,50) (numerically) with a relative accuracy of 1 part in 10000000. Define the function. N is a counter, so we can see how many function evaluations were needed. F(X):=(MODE_DECLARE(N,INTEGER,X,FLOAT),N:N+1,EXP(-X))$ TRANSLATE(F)$ /* First of all try doing the whole integral at once */ BLOCK([ROMBERGTOL:1.E-6,ROMBERABS:0.],N:0,ROMBERG(F,0,50)); ==> 1.00000003 N; ==> 257 /* Number of function evaluations*/ Now do the integral intelligently, by first doing Integral(exp(-x),x,0,10) and then setting ROMBERGABS to 1.E-6*(this partial integral). BLOCK([ROMBERGTOL:1.E-6,ROMBERGABS:0.,SUM:0.], N:0,SUM:ROMBERG(F,0,10),ROMBERGABS:SUM*ROMBERGTOL,ROMBERGTOL:0., SUM+ROMBERG(F,10,50)); ==> 1.00000001 /* Same as before */ N; ==> 130 So if F(X) were a function that took a long time to compute, the second method would be about 2 times quicker. - Variable: ROMBERGIT default: [11] - The accuracy of the ROMBERG integration command is governed by the global variables ROMBERGTOL[1.E-4] and ROMBERGIT[11]. ROMBERG will return a result if the relative difference in successive approximations is less than ROMBERGTOL. It will try halving the stepsize ROMBERGIT times before it gives up. - Variable: ROMBERGMIN default: [0] - governs the minimum number of function evaluations that ROMBERG will make. ROMBERG will evaluate its first arg. at least 2^(ROMBERGMIN+2)+1 times. This is useful for integrating oscillatory functions, when the normal converge test might sometimes wrongly pass. - Variable: ROMBERGTOL default: [1.E-4] - The accuracy of the ROMBERG integration command is governed by the global variables ROMBERGTOL[1.E-4] and ROMBERGIT[11]. ROMBERG will return a result if the relative difference in successive approximations is less than ROMBERGTOL. It will try halving the stepsize ROMBERGIT times before it gives up. - Function: TLDEFINT (exp,var,ll,ul) is just LDEFINT with TLIMSWITCH set to TRUE. File: maxima.info, Node: Equations, Next: Differential Equations, Prev: Integration, Up: Top Equations ********* * Menu: * Definitions for Equations:: File: maxima.info, Node: Definitions for Equations, Prev: Equations, Up: Equations Definitions for Equations ========================= - Variable: %RNUM_LIST default: [] - When %R variables are introduced in solutions by the ALGSYS command, they are added to %RNUM_LIST in the order they are created. This is convenient for doing substitutions into the solution later on. It's recommended to use this list rather than doing CONCAT('%R,J). - Variable: ALGEXACT default: [FALSE] affects the behavior of ALGSYS as follows: If ALGEXACT is TRUE, ALGSYS always calls SOLVE and then uses REALROOTS on SOLVE's failures. If ALGEXACT is FALSE, SOLVE is called only if the eliminant was not univariate, or if it was a quadratic or biquadratic. Thus ALGEXACT:TRUE doesn't guarantee only exact solutions, just that ALGSYS will first try as hard as it can to give exact solutions, and only yield approximations when all else fails. - Function: ALGSYS ([exp1, exp2, ...], [var1, var2, ...]) solves the list of simultaneous polynomials or polynomial equations (which can be non-linear) for the list of variables. The symbols %R1, %R2, etc. will be used to represent arbitrary parameters when needed for the solution (the variable %RNUM_LIST holds these). In the process described below, ALGSYS is entered recursively if necessary. The method is as follows: (1) First the equations are FACTORed and split into subsystems. (2) For each subsystem Si, an equation E and a variable var are selected (the var is chosen to have lowest nonzero degree). Then the resultant of E and Ej with respect to var is computed for each of the remaining equations Ej in the subsystem Si. This yields a new subsystem S'i in one fewer variables (var has been eliminated). The process now returns to (1). (3) Eventually, a subsystem consisting of a single equation is obtained. If the equation is multivariate and no approximations in the form of floating point numbers have been introduced, then SOLVE is called to find an exact solution. (The user should realize that SOLVE may not be able to produce a solution or if it does the solution may be a very large expression.) If the equation is univariate and is either linear, quadratic, or bi-quadratic, then again SOLVE is called if no approximations have been introduced. If approximations have been introduced or the equation is not univariate and neither linear, quadratic, or bi-quadratic, then if the switch REALONLY[FALSE] is TRUE, the function REALROOTS is called to find the real-valued solutions. If REALONLY:FALSE then ALLROOTS is called which looks for real and complex-valued solutions. If ALGSYS produces a solution which has fewer significant digits than required, the user can change the value of ALGEPSILON[10^8] to a higher value. If ALGEXACT[FALSE] is set to TRUE, SOLVE will always be called. (4) Finally, the solutions obtained in step (3) are re-inserted into previous levels and the solution process returns to (1). The user should be aware of several caveats: When ALGSYS encounters a multivariate equation which contains floating point approximations (usually due to its failing to find exact solutions at an earlier stage), then it does not attempt to apply exact methods to such equations and instead prints the message: "ALGSYS cannot solve - system too complicated." Interactions with RADCAN can produce large or complicated expressions. In that case, the user may use PICKAPART or REVEAL to analyze the solution. Occasionally, RADCAN may introduce an apparent %I into a solution which is actually real-valued. Do EXAMPLE(ALGSYS); for examples. - Function: ALLROOTS (poly) finds all the real and complex roots of the real polynomial poly which must be univariate and may be an equation, e.g. poly=0. For complex polynomials an algorithm by Jenkins and Traub is used (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97). For real polynomials the algorithm used is due to Jenkins (Algorithm 493, TOMS, vol. 1, (1975), p.178). The flag POLYFACTOR[FALSE] when true causes ALLROOTS to factor the polynomial over the real numbers if the polynomial is real, or over the complex numbers, if the polynomial is complex. ALLROOTS may give inaccurate results in case of multiple roots. (If poly is real and you get inaccurate answers, you may want to try ALLROOTS(%I*poly);) Do EXAMPLE(ALLROOTS); for an example. ALLROOTS rejects non-polynomials. It requires that the numerator after RATting should be a polynomial, and it requires that the denominator be at most a complex number. As a result of this ALLROOTS will always return an equivalent (but factored) expression, if POLYFACTOR is TRUE. - Variable: BACKSUBST default: [TRUE] if set to FALSE will prevent back substitution after the equations have been triangularized. This may be necessary in very big problems where back substitution would cause the generation of extremely large expressions. (On MC this could cause storage capacity to be exceeded.) - Variable: BREAKUP default: [TRUE] if FALSE will cause SOLVE to express the solutions of cubic or quartic equations as single expressions rather than as made up of several common subexpressions which is the default. BREAKUP:TRUE only works when PROGRAMMODE is FALSE. - Function: DIMENSION (equation or list of equations) The file "share1/dimen.mc" contains functions for automatic dimensional analysis. LOAD(DIMEN); will load it up for you. There is a demonstration available in share1/dimen.dem. Do DEMO("dimen"); to run it. - Variable: DISPFLAG default: [TRUE] if set to FALSE within a BLOCK will inhibit the display of output generated by the solve functions called from within the BLOCK. Termination of the BLOCK with a dollar sign, $, sets DISPFLAG to FALSE. - Function: FUNCSOLVE (eqn,g(t)) gives [g(t) = ...] or [], depending on whether or not there exists a rational fcn g(t) satisfying eqn, which must be a first order, linear polynomial in (for this case) g(t) and g(t+1). (C1) FUNCSOLVE((N+1)*FOO(N)-(N+3)*FOO(N+1)/(N+1) = (N-1)/(N+2),FOO(N)); N (D1) FOO(N) = --------------- (N + 1) (N + 2) Warning: this is a very rudimentary implementation-many safety checks and obvious generalizations are missing. - Variable: GLOBALSOLVE default: [FALSE] if set to TRUE then variables which are SOLVEd for will be set to the solution of the set of simultaneous equations. - Function: IEQN (ie,unk,tech,n,guess) Integral Equation solving routine. Do LOAD(INTEQN); to access it. CAVEAT: To free some storage, a KILL(LABELS) is included in this file. Therefore, before loading the integral equation package, the user should give names to any expressions he wants to keep. ie is the integral equation; unk is the unknown function; tech is the technique to be tried from those given above (tech = FIRST means: try the first technique which finds a solution; tech = ALL means: try all applicable techniques); n is the maximum number of terms to take for TAYLOR, NEUMANN, FIRSTKINDSERIES, or FREDSERIES (it is also the maximum depth of recursion for the differentiation method); guess is the initial guess for NEUMANN or FIRSTKINDSERIES. Default values for the 2nd thru 5th parameters are: unk: P(X), where P is the first function encountered in an integrand which is unknown to MACSYMA and X is the variable which occurs as an argument to the first occurrence of P found outside of an integral in the case of SECONDKIND equations, or is the only other variable besides the variable of integration in FIRSTKIND equations. If the attempt to search for X fails, the user will be asked to supply the independent variable; tech: FIRST; n: 1; guess: NONE, which will cause NEUMANN and FIRSTKINDSERIES to use F(X) as an initial guess. - Variable: IEQNPRINT default: [TRUE] - governs the behavior of the result returned by the IEQN command (which see). If IEQNPRINT is set to FALSE, the lists returned by the IEQN function are of the form [SOLUTION, TECHNIQUE USED, NTERMS, FLAG] where FLAG is absent if the solution is exact. Otherwise, it is the word APPROXIMATE or INCOMPLETE corresponding to an inexact or non-closed form solution, respectively. If a series method was used, NTERMS gives the number of terms taken (which could be less than the n given to IEQN if an error prevented generation of further terms). - Function: LHS (eqn) the left side of the equation eqn. - Function: LINSOLVE ([exp1, exp2, ...], [var1, var2, ...]) solves the list of simultaneous linear equations for the list of variables. The expi must each be polynomials in the variables and may be equations. If GLOBALSOLVE[FALSE] is set to TRUE then variables which are SOLVEd for will be set to the solution of the set of simultaneous equations. BACKSUBST[TRUE] if set to FALSE will prevent back substitution after the equations have been triangularized. This may be necessary in very big problems where back substitution would cause the generation of extremely large expressions. (On MC this could cause the storage capacity to be exceeded.) LINSOLVE_PARAMS[TRUE] If TRUE, LINSOLVE also generates the %Ri symbols used to represent arbitrary parameters described in the manual under ALGSYS. If FALSE, LINSOLVE behaves as before, i.e. when it meets up with an under-determined system of equations, it solves for some of the variables in terms of others. (C1) X+Z=Y$ (C2) 2*A*X-Y=2*A**2$ (C3) Y-2*Z=2$ (C4) LINSOLVE([D1,D2,D3],[X,Y,Z]),GLOBALSOLVE:TRUE; SOLUTION (E4) X : A + 1 (E5) Y : 2 A (E6) Z : A - 1 (D6) [E4, E5, E6] - Variable: LINSOLVEWARN default: [TRUE] - if FALSE will cause the message "Dependent equations eliminated" to be suppressed. - Variable: LINSOLVE_PARAMS default: [TRUE] - If TRUE, LINSOLVE also generates the %Ri symbols used to represent arbitrary parameters described in the manual under ALGSYS. If FALSE, LINSOLVE behaves as before, i.e. when it meets up with an under-determined system of equations, it solves for some of the variables in terms of others. - Variable: MULTIPLICITIES default: [NOT_SET_YET] - will be set to a list of the multiplicities of the individual solutions returned by SOLVE or REALROOTS. - Function: NROOTS (poly, low, high) finds the number of real roots of the real univariate polynomial poly in the half-open interval (low,high]. The endpoints of the interval may also be MINF,INF respectively for minus infinity and plus infinity. The method of Sturm sequences is used. (C1) POLY1:X**10-2*X**4+1/2$ (C2) NROOTS(POLY1,-6,9.1); RAT REPLACED 0.5 BY 1/2 = 0.5 (D2) 4 - Function: NTHROOT (p,n) where p is a polynomial with integer coefficients and n is a positive integer returns q, a polynomial over the integers, such that q^n=p or prints an error message indicating that p is not a perfect nth power. This routine is much faster than FACTOR or even SQFR. - Variable: PROGRAMMODE default: [TRUE] - when FALSE will cause SOLVE, REALROOTS, ALLROOTS, and LINSOLVE to print E-labels (intermediate line labels) to label answers. When TRUE, SOLVE, etc. return answers as elements in a list. (Except when BACKSUBST is set to FALSE, in which case PROGRAMMODE:FALSE is also used.) - Variable: REALONLY default: [FALSE] - if TRUE causes ALGSYS to return only those solutions which are free of %I. - Function: REALROOTS (poly, bound) finds all of the real roots of the real univariate polynomial poly within a tolerance of bound which, if less than 1, causes all integral roots to be found exactly. The parameter bound may be arbitrarily small in order to achieve any desired accuracy. The first argument may also be an equation. REALROOTS sets MULTIPLICITIES, useful in case of multiple roots. REALROOTS(poly) is equivalent to REALROOTS(poly,ROOTSEPSILON). ROOTSEPSILON[1.0E-7] is a real number used to establish the confidence interval for the roots. Do EXAMPLE(REALROOTS); for an example. - Function: RHS (eqn) the right side of the equation eqn. - Variable: ROOTSCONMODE default: [TRUE] - Determines the behavior of the ROOTSCONTRACT command. Do DESCRIBE(ROOTSCONTRACT); for details. - Function: ROOTSCONTRACT (exp) converts products of roots into roots of products. For example, ROOTSCONTRACT(SQRT(X)*Y^(3/2)) ==> SQRT(X*Y^3) When RADEXPAND is TRUE and DOMAIN is REAL (their defaults), ROOTSCONTRACT converts ABS into SQRT, e.g. ROOTSCONTRACT(ABS(X)*SQRT(Y)) ==> SQRT(X^2*Y) There is an option ROOTSCONMODE (default value TRUE), affecting ROOTSCONTRACT as follows: Problem Value of Result of applying ROOTSCONMODE ROOTSCONTRACT X^(1/2)*Y^(3/2) FALSE (X*Y^3)^(1/2) X^(1/2)*Y^(1/4) FALSE X^(1/2)*Y^(1/4) X^(1/2)*Y^(1/4) TRUE (X*Y^(1/2))^(1/2) X^(1/2)*Y^(1/3) TRUE X^(1/2)*Y^(1/3) X^(1/2)*Y^(1/4) ALL (X^2*Y)^(1/4) X^(1/2)*Y^(1/3) ALL (X^3*Y^2)^(1/6) The above examples and more may be tried out by typing EXAMPLE(ROOTSCONTRACT); When ROOTSCONMODE is FALSE, ROOTSCONTRACT contracts only wrt rational number exponents whose denominators are the same. The key to the ROOTSCONMODE:TRUE$ examples is simply that 2 divides into 4 but not into 3. ROOTSCONMODE:ALL$ involves taking the lcm (least common multiple) of the denominators of the exponents. ROOTSCONTRACT uses RATSIMP in a manner similar to LOGCONTRACT (see the manual). - Variable: ROOTSEPSILON default: [1.0E-7] - a real number used to establish the confidence interval for the roots found by the REALROOTS function. - Function: SOLVE (exp, var) solves the algebraic equation exp for the variable var and returns a list of solution equations in var. If exp is not an equation, it is assumed to be an expression to be set equal to zero. Var may be a function (e.g. F(X)), or other non-atomic expression except a sum or product. It may be omitted if exp contains only one variable. Exp may be a rational expression, and may contain trigonometric functions, exponentials, etc. The following method is used: Let E be the expression and X be the variable. If E is linear in X then it is trivially solved for X. Otherwise if E is of the form A*X**N+B then the result is (-B/A)**(1/N) times the Nth roots of unity. If E is not linear in X then the gcd of the exponents of X in E (say N) is divided into the exponents and the multiplicity of the roots is multiplied by N. Then SOLVE is called again on the result. If E factors then SOLVE is called on each of the factors. Finally SOLVE will use the quadratic, cubic, or quartic formulas where necessary. In the case where E is a polynomial in some function of the variable to be solved for, say F(X), then it is first solved for F(X) (call the result C), then the equation F(X)=C can be solved for X provided the inverse of the function F is known. BREAKUP[TRUE] if FALSE will cause SOLVE to express the solutions of cubic or quartic equations as single expressions rather than as made up of several common subexpressions which is the default. MULTIPLICITIES[NOT_SET_YET] - will be set to a list of the multiplicities of the individual solutions returned by SOLVE, REALROOTS, or ALLROOTS. Try APROPOS(SOLVE) for the switches which affect SOLVE. DESCRIBE may then by used on the individual switch names if their purpose is not clear. SOLVE([eq1, ..., eqn], [v1, ..., vn]) solves a system of simultaneous (linear or non-linear) polynomial equations by calling LINSOLVE or ALGSYS and returns a list of the solution lists in the variables. In the case of LINSOLVE this list would contain a single list of solutions. It takes two lists as arguments. The first list (eqi, i=1,...,n) represents the equations to be solved; the second list is a list of the unknowns to be determined. If the total number of variables in the equations is equal to the number of equations, the second argument-list may be omitted. For linear systems if the given equations are not compatible, the message INCONSISTENT will be displayed (see the SOLVE_INCONSISTENT_ERROR switch); if no unique solution exists, then SINGULAR will be displayed. For examples, do EXAMPLE(SOLVE); - Variable: SOLVEDECOMPOSES default: [TRUE] - if TRUE, will induce SOLVE to use POLYDECOMP (see POLYDECOMP) in attempting to solve polynomials. - Variable: SOLVEEXPLICIT default: [FALSE] - if TRUE, inhibits SOLVE from returning implicit solutions i.e. of the form F(x)=0. - Variable: SOLVEFACTORS default: [TRUE] - if FALSE then SOLVE will not try to factor the expression. The FALSE setting may be desired in some cases where factoring is not necessary. - Variable: SOLVENULLWARN default: [TRUE] - if TRUE the user will be warned if he calls SOLVE with either a null equation list or a null variable list. For example, SOLVE([],[]); would print two warning messages and return []. - Variable: SOLVERADCAN default: [FALSE] - if TRUE then SOLVE will use RADCAN which will make SOLVE slower but will allow certain problems containing exponentials and logs to be solved. - Variable: SOLVETRIGWARN default: [TRUE] - if set to FALSE will inhibit printing by SOLVE of the warning message saying that it is using inverse trigonometric functions to solve the equation, and thereby losing solutions. - Variable: SOLVE_INCONSISTENT_ERROR default: [TRUE] - If TRUE, SOLVE and LINSOLVE give an error if they meet up with a set of inconsistent linear equations, e.g. SOLVE([A+B=1,A+B=2]). If FALSE, they return [] in this case. (This is the new mode, previously gotten only by calling ALGSYS.) - Function: ZRPOLY - This is no longer available in Maxima. See ALLROOTS for a function to compute the roots of a polynomial. - Function: ZSOLVE This is not available with Maxima anymore. Documentation is left for historical purposes. - For those who can make use of approximate numerical solutions to problems, there is a package which calls a routine which has been translated from the IMSL fortran library to solve N simultaneous non-linear equations in N unknowns. It uses black-box techniques that probably aren't desirable if an exact solution can be obtained from one of the smarter solvers (LINSOLVE, ALGSYS, etc). But for things that the other solvers don't attempt to handle, this can probably give some very useful results. For documentation, do PRINTFILE("zsolve.usg");. For a demo do batch("zsolve.mc")$ File: maxima.info, Node: Differential Equations, Next: Numerical, Prev: Equations, Up: Top Differential Equations ********************** * Menu: * Definitions for Differential Equations::