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Package roots (in roots.i) -

Index of documented functions or symbols:

f_inverse

mnbrent

mxbrent

nraphson

roots

### f_inverse

```DOCUMENT f_inverse(f_and_dfdx, y, x0, x1, xerr)
or f_inverse(f_and_dfdx, y, x0, x1, xerr)

Find values of an inverse function by Newton-Raphson iteration,
backed up by bisection if the convergence seems poor.  The
subroutine F_AND_DFDX must be defined as:
func F_AND_DFDX
returning both the function value f(x) and derivative dfdx(x).
If the input x is an array, the returned f and dfdx must have
the same shape as the input x.  If F_AND_DFDX always returns
zero dfdx, f_inverse will use bisection.

The result x will have the same shape as the input Y values.

The values of x are constrained to lie within the interval from
X0 to X1; the function value must be on opposite sides of the
required Y at these interval endpoints.  The iteration stops
when the root is known to within XERR, or to machine precision
if XERR is nil or zero.  X0, X1, and XERR may be arrays conformable
with Y.

f_inverse takes the same number of iterations for every Y value;
it does not notice that some may have converged before others.
```

### mnbrent

```DOCUMENT fmin= mnbrent(f, x0, x1, x2)
or fmin= mnbrent(f, x0, x1, x2, xmin)
or fmin= mnbrent(f, x0, x1, x2, xmin, xerr)

returns the minimum of the function F (of a single argument x),
given three points X0, X1, and X2 such that F(X1) is less than
either F(X0) or F(X2), and X1 is between X0 and X2.  If the
XMIN argument is provided, it is set to the x value which
produced FMIN.  If XERR is supplied, the search stops when
a fractional error of XERR in x is reached; note that XERR
smaller than the square root of the machine precision (or
omitted) will cause convergence to machine precision in FMIN.

The algorithm is Brent's method - a combination of inverse
parabolic interpolation and golden section search - as adapted
from Numerical Recipes Ch. 10 (Press, et. al.).
```

### mxbrent

```DOCUMENT fmax= mxbrent(f, x0, x1, x2)
or fmax= mxbrent(f, x0, x1, x2, xmax)
or fmax= mxbrent(f, x0, x1, x2, xmax, xerr)

returns the maximum of the function F (of a single argument x),
given three points X0, X1, and X2 such that F(X1) is greater than
either F(X0) or F(X2), and X1 is between X0 and X2.  If the
XMAX argument is provided, it is set to the x value which
produced FMAX.  If XERR is supplied, the search stops when
a fractional error of XERR in x is reached; note that XERR
smaller than the square root of the machine precision (or
omitted) will cause convergence to machine precision in FMAX.

The algorithm is Brent's method - a combination of inverse
parabolic interpolation and golden section search - as adapted
from Numerical Recipes Ch. 10 (Press, et. al.).
```

### nraphson

```DOCUMENT nraphson(f_and_dfdx, x0, x1)
or nraphson(f_and_dfdx, x0, x1, xerr)

Find a root of a function by Newton-Raphson iteration, backed
up by bisection if the convergence seems poor.  The subroutine
F_AND_DFDX must be defined as:
func F_AND_DFDX
returning both the function value f(x) and derivative dfdx(x).
If F_AND_DFDX always returns dfdx==0, nraphson uses bisection.
The value of x is constrained to lie within the interval from
X0 to X1; the function values at these two points must have
opposite sign.  The iteration stops when the root is known to
within XERR, or to machine precision if XERR is nil or zero.

f_inverse is a "vectorized" version of nraphson.

Based on rtsafe from Press, et. al. Numerical Recipes, Ch 9.
```

```DOCUMENT roots.i