ADAMS VARIABLE STEP-SIZE PREDICTOR-CORRECTOR ALGORITHM
*
* To approximate the solution of the initial value problem
*
* y' = f( t, y ), a <= t <= b, y(a) = ALPHA,
*
* with local truncation error within a given tolerance:
*
* INPUT: endpoints a, b; initial condition ALPHA; tolerance TOL;
* maximum step size HMAX; minimum step size HMIN.
*
* OUTPUT: I, T(I), W(I), H where at the Ith step W(I) approximates
* y(T(I)) and step size H was used or a message that the
* minimum step size was exceeded.
ADAMS-FORTH ORDER PREDICTOR-CORRECTOR ALGORITHM
*
* To approximate the solution of the initial value problem
* y' = f(t,y), a <= t <= b, y(a) = alpha,
* at N+1 equally spaced points in the interval [a,b].
*
* INPUT: endpoints a,b; initial condition alpha; integer N.
*
* OUTPUT: approximation w to y at the (N+1) values of t.
EULER'S ALGORITHM
*
* TO APPROXIMATE THE SOLUTION OF THE INITIAL VALUE PROBLEM:
* Y' = F(T,Y), A<=T<=B, Y(A) = ALPHA,
* AT N+1 EQUALLY SPACED POINTS IN THE INTERVAL [A,B].
*
* INPUT: ENDPOINTS A,B; INITIAL CONDITION ALPHA; INTEGER N.
*
* OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T.
EXTRAPOLATION ALGORITHM
*
* To approximate the solution of the initial value problem:
* y' = f(t,y), a <= t <= b, y(a) = ALPHA,
* with local truncation error within a given tolerance:
*
* INPUT: endpoints a,b; initial condition ALPHA; tolerance TOL;
* maximum stepsize HMAX; minimum stepsize HMIN.
*
* OUTPUT: T, W, H where W approximates y(T) and stepsize H was
* used or a message that minimum stepsize was exceeded.
RUNGE-KUTTA (ORDER
4) ALGORITHM
*
* TO APPROXIMATE THE SOLUTION TO THE INITIAL VALUE PROBLEM:
* Y' = F(T,Y), A<=T<=B, Y(A) = ALPHA,
* AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].
*
* INPUT: ENDPOINTS A,B; INITIAL CONDITION ALPHA; INTEGER N.
*
* OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T.
RUNGE-KUTTA
FOR SYSTEMS OF DIFFERENTIAL EQUATIONS ALGORITHM
*
* TO APPROXIMATE THE SOLUTION OF THE MTH-ORDER SYSTEM OF FIRST-
* ORDER INITIAL-VALUE PROBLEMS
* UJ' = FJ( T, U1, U2, ..., UM ), J = 1, 2, ..., M
* A <= T <= B, UJ(A) = ALPHAJ, J = 1, 2, ..., M
* AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].
*
* INPUT: ENDPOINTS A,B; NUMBER OF EQUATIONS M; INITIAL
* CONDITIONS ALPHA1, ..., ALPHAM; INTEGER N.
*
* OUTPUT: APPROXIMATION WJ TO UJ(T) AT THE (N+1) VALUES OF T.
RUNGE-KUTTA-FEHLBERG ALGORITHM
*
* TO APPROXIMATE THE SOLUTION OF THE INITIAL VALUE PROBLEM:
* Y' = F(T,Y), A<=T<=B, Y(A) = ALPHA,
* WITH LOCAL TRUNCATION ERROR WITHIN A GIVEN TOLERANCE.
*
* INPUT: ENDPOINTS A,B; INITIAL CONDITION ALPHA; TOLERANCE TOL;
* MAXIMUM STEPSIZE HMAX; MINIMUM STEPSIZE HMIN.
*
* OUTPUT: T, W, H WHERE W APPROXIMATES Y(T) AND STEPSIZE H WAS
* USED OR A MESSAGE THAT MINIMUM STEPSIZE WAS EXCEEDED.
TRAPEZOIDAL
WITH NEWTON ITERATION ALGORITHM
*
* TO APPROXIMATE THE SOLUTION OF THE INITIAL VALUE PROBLEM:
* Y' = F(T,Y), A <= T <= B, Y(A) = ALPHA,
* AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].
*
* INPUT: ENDPOINTS A,B; INITIAL CONDITION ALPHA; INTEGER N;
* TOLERANCE TOL; MAXIMUM NUMBER OF ITERATIONS M AT ANY ONE STEP.
*
* OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T
* OR A MESSAGE OF FAILURE.