ROOTS

The Newton method can be used to find the solutions of

f(x)=0 

Suppose x₀ is a sufficiently close approximation to a solution.  Then a better approximation x₁is found by:

                   f(x₀)
x₁= x₀ -  ¾¾¾
                  f ’(x₀)

The specified interval is explored for changes in the sign of the function.  If one is detected in a given subinterval, then a root must lie inside that subinterval. The center of it is then chosen as a first approximation and Newton's method applied to obtain the sequence of approximations:  x₀, x₁, x₂, ... , x_n.  The x_n is assumed correct if it differs from the previous value in the sequence by less than the specified error.

SYSTEMS OF SIMULTANEOUS NONLINEAR EQUATIONS

Suppose that, instead of a single nonlinear equation, we have a systems system of simultaneous nonlinear equations. We can solve it using the generalized Newton method, which is just a generalization of what we did to find the roots of a one-variable function. To illustrate, suppose we want to solve the system:

F(x,y,z) = 0  ;  G(x,y,z) = 0  ;  H(x,y,z) = 0         (I)

Assume (x₀,y₀,z₀) is close to a solution.  Now expand all three equations into their Taylor series.  Neglecting terms of order higher than the first, we get:

F_x*(x-x₀) + F_y*(y-y₀) + F_z*(z-z₀) = - F(x₀,y₀,z₀)
G_x*(x-x₀) + G_y*(y-y₀) + G_z*(z-z₀) = - G(x₀,y₀,z₀)          (II)
H_x*(x-x₀) + H_y*(y-y₀) + H_z*(z-z₀) = - H(x₀,y₀,z₀)

            ¶F        ¶F         ¶F
where  F_x = ———, F_y = ———— , F_z = ——— , all evaluated at (x₀,y₀,z₀).
            ¶x        ¶y         ¶z

Similarly for G_x, G_y, G_z, H_x, H_y, H_z.

Solving system (II) for (x-x₀), (y- y₀) and (z-z₀), we obtain corrections on the initial approximation, that is, if its solutions are D_x, D_y and D_z, where

x-x₀=D_x;  y-y₀=D_y ;  z-z₀=D_z

then a closer approximation to the solution is

x₁=x₀+D_x ;  y₁=y₀+D_y ;  z₁=z₀+D_z

We now use x₁, y₁ and z₁ as initial approximations and again set a system of linear equations then solve it to obtain x₂, y₂, and z₂.  This process is continued until all the variables in the current approximation in the sequence differ from the values of the previous approximation by less than the specified error and all equations differ from zero by less than said error.

funactions 

Available Functions

In addition to the functions generally available, this program lets you use 8 additional mathematical functions. These are sinh, cosh, tanh, asinh, acosh, atanh, asin, acos. They return, correspondingly, the hyperbolic sine, cosine, and tangent, inverse hyperbolic sine, cosine, and tangent, inverse trigonometric sine and cosine.
Below is a list of the functions available, including the 8 that have been added.
FUNCTIONS:     RETURNS:
sgn            sign
int            integer value (fractions are discarded)
cint           integer value (fractions are rounded)
abs            absolute value
rnd            random
sqr            square root
log            natural logarithm
exp            exponential
sin            trigonometric sine
cos            trigonometric cosine
tan            trigonometric tangent
asin           inverse trigonometric sine
acos           inverse trigonometric cosine
atn            inverse trigonometric tangent
sinh           hyperbolic sine
cosh           hyperbolic cosine
tanh           hyperbolic tangent
asinh          inverse hyperbolic sine
acosh          inverse hyperbolic cosine
atanh          inverse hyperbolic tangent

Available Operators

Arithmetic Operators (In order of precedence)
Negation (-)
Exponentiation (^)
Multiplication and division (*, /)
Integer division (\)
Modulus arithmetic (mod)
Addition and subtraction (+, -)
Comparison Operators (These all have EQUAL precedence)
Equality (=)
Inequality (<>)
Less than (<)
Greater than (>)
Less than or equal to (<=)
Greater than or equal to (>=)
Logical Operators (In order of precedence)
Negation (not)
Conjunction (and)
Disjunction (or)
Exclusion (xor)
Equivalence (eqv)
Implication (imp)
Notes:
If an expression contains operators from different categories, the Arithmetic Operators are evaluated first, then the Comparison Operators, then the Logical Operators. For each category, the precedence is as listed above.
When operators of equal precedence are found, they are evaluated in order of appearance from left to right.
Parentheses always override Operator Precedence rules. Expressions are evaluated from the inside out, starting with the innermost set of parentheses. Use parentheses to eliminate any possible ambiguity.
You’ll probably use the Arithmetic Operators more often than the others.
Built-in Constants
The constants pi and e are built-in for your convenience:
pi = 3.14159… = atn(1)*4