function [ xmin, ymin ] = minabs ( x1, y1, x2, y2, x3, y3 )
%*****************************************************************************80
%
%% minabs() finds a local minimum of F(X) = A * abs ( X ) + B.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 08 March 2005
%
% Author:
%
% John Burkardt
%
% Input:
%
% real X1, Y1, X2, Y2, X3, Y3, are three sets of
% data of the form ( X, F(X) ). The three X values must be distinct.
%
% Output:
%
% real XMIN, YMIN. XMIN is a point within the interval
% spanned by X1, X2 and X3, at which F takes its local minimum
% value YMIN.
%
%
% Refuse to deal with coincident data.
%
if ( x1 == x2 || x2 == x3 || x3 == x1 )
fprintf ( 1, '\n' );
fprintf ( 1, 'MINABS - Fatal error!\n' );
fprintf ( 1, ' X values are equal.\n' );
error ( 'MINABS - Fatal error!' );
end
%
% Sort the data.
%
if ( x2 < x1 )
t = x1;
x1 = x2;
x2 = t;
t = y1;
y1 = y2;
y2 = t;
end
if ( x3 < x1 )
t = x1;
x1 = x3;
x3 = t;
t = y1;
y1 = y3;
y3 = t;
end
if ( x3 < x2 )
t = x2;
x2 = x3;
x3 = t;
t = y2;
y2 = y3;
y3 = t;
end
%
% Now determine the slopes.
%
slope12 = ( y2 - y1 ) / ( x2 - x1 );
slope23 = ( y3 - y2 ) / ( x3 - x2 );
slope13 = ( y3 - y1 ) / ( x3 - x1 );
%
% Case 1: Minimum must be at an endpoint.
%
if ( slope13 <= slope12 || 0.0 <= slope12 )
if ( y1 < y3 )
xmin = x1;
ymin = y1;
else
xmin = x3;
ymin = y3;
end
%
% Case 2: The curve decreases, and decreases faster than the line
% joining the endpoints.
%
% Whichever of SLOPE12 and SLOPE23 is the greater in magnitude
% represents the actual slope of the underlying function.
% Find where two lines of that slope, passing through the
% endpoint data, intersect.
%
else
slope = max ( abs ( slope12 ), slope23 );
xmin = 0.5 * ( x1 + x3 + ( y1 - y3 ) / slope );
ymin = y1 - slope * ( xmin - x1 );
end
return
end