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# Integration

## Introduction to Integration

MACSYMA has several routines for handling integration. The INTEGRATE command makes use of most of them. There is also the ANTID package, which handles an unspecified function (and its derivatives, of course). For numerical uses, there is the ROMBERG function. There is also an adaptave integrator which uses the Newton-Cotes 8 panel quadrature rule, called QUANC8. Hypergeometric Functions are being worked on, do DESCRIBE(SPECINT); for details. Generally speaking, MACSYMA only handles integrals which are integrable in terms of the "elementary functions" (rational functions, trigonometrics, logs, exponentials, radicals, etc.) and a few extensions (error function, dilogarithm). It does not handle integrals in terms of unknown functions such as g(x) and h(x).

## 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.

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