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Minimization Along A Direction In A Vector Space
Syntax [betaout, nf] = minline(function f,
xref, betain, dir, level, maxdx, maxf)
See Also brent , conjdir , conjgrad

Description
Sets betaout the value of beta that solves the problem:
     minimize f(xref + beta dir) with respect to beta
where beta is a scalar with the same type as xref. If minline can not decrease the value of f in the direction dir, betaout is exactly equal to zero. If present, the return value nf is set to the number of function evaluations used for the minimization.

f(x)
returns a scalar value with the same type as xref provided that x has the same type and dimension as xref.

xref
is a real or double-precision column vector that specifies the reference point for the line search.

betain
is a nonzero scalar that has the same type as xref and specifies the initial step size for the line search parameter.

dir
is a vector with the same type and dimension as xref. It specifies the direction of the line search.

level
is an integer scalar that specifies the level of tracing inside minline.
Name Condition Description
nf level > 1 number calls to f
beta level > 1 step factor
f(x) level > 1 objective value
bmin level > 1 final step
fmin level > 1 final objective


maxdx
if present, maxdx specifies the maximum absolute change; that is,

     | beta dir  |  < maxdx  for all i
               i           i
and maxdx must have the same type and dimension as xref. If maxdx is not present, the vector defined by
     maxdx  = 100 betain (|dir | + |dir| / 100)
          i                   i
is used.

nf
if present, maxf specifies the maximum number of evaluations of f to allow. If minline terminates because of this limit, nf is set to maxf + 1

Example
The following example minimizes the function
                       2                      2
     (2 + beta * 0 - 1)  +  (2 + beta * 1 - 1)

with respect to beta.
     clear
     function f(x) begin
               return (x(1) - 1.)^2 + (x(2) - 1.)^2
     end
     xref   = {2., 2.}
     betain = 1 / 3.
     dir    = {0., 1.}
     minline(function f, xref, betain, dir)
 returns
     -1