function scoreout = pivotgolf(course,pivotstrat)
% PIVOTGOLF Pivot Pickin' Golf.
% Your goal is to use LUGUI to compute the LU decompositions of nine
% matrices with as little roundoff error as possible. Your score for
% each hole is norm(R,inf)+norm(Leps,inf)+norm(Ueps,inf) where
% R = L*U - A(p,q) is the residual and Leps and Ueps are the nonzeros
% that should be zero in L and U.
%
% The six golf courses include:
% magic -- magic squares and Golub matrices, some are rank deficient.
% testmats -- various test matrices, Pascal, Hilbert, gallery, etc.
% rand #s -- random integer matrices initialized by rand('state',s).
%
% The four pivot strategies are:
% pick -- use the mouse to pick the pivots.
% diagonal -- pivot on the diagonal, it is possible to divide by zero.
% partial -- pivot is the largest element in the current column.
% complete -- pivot is the largest element in the unreduced matrix.
%
% PIVOTGOLF(course,pivotstrat) bypasses the initial screen.
% course = 'magic', 'testmats', or a numeric s to set rand('state',s).
% pivotstrat = 'pick','diagonal','partial', or 'complete'.
%
% Try to pick pivot elements that divide into the other elements exactly.
% If you can choose a pivot element that is a power of two, there will
% be no roundoff error at that step. But if the pivot is small, the
% elements in the unreduced matrix might grow larger and subsequent steps
% might have larger roundoff errors.
%
% You should know that "mulligan" is golf terminology for "OOPS, I want
% to take that shot over". Good luck!
%
% See also LUGUI.
if nargin == 0
% Initial screen
shg
clf
axes('pos',[0 0 1 1]);
axis off
set(gcf,'double','on','name','Pivot Pickin'' Golf', ...
'menu','none','numbertitle','off','color','white')
text('units','norm','pos',[.25,.75],'color',[0 0.65 0], ...
'fontweight','bold','fontsize',16, ...
'string',sprintf('Welcome to Pivot Pickin'' Golf'))
text('units','norm','pos',[.03 .60],'fontweight','bold','string','Course:')
for k = 1:6
switch k
case 1,
str = 'magic';
case 2,
str = 'testmats';
otherwise,
str = ['rand #' int2str(k-2)];
end
b(k) = uicontrol('units','norm','pos',[.14*k .57 .12 .06], ...
'style','toggle','string',str,'back','w');
end
text('units','norm','pos',[.03 .45],'fontweight','bold','string','Strategy:')
str = {'pick','diagonal','partial','complete'};
for k = 1:4
b(k+6) = uicontrol('units','norm','pos',[.14*k .42 .12 .06], ...
'style','toggle','string',str{k},'back','w');
end
stop = uicontrol('style','toggle','string','X','fontweight','bold', ...
'back','white','units','norm','pos',[.96 .96 .04 .04]);
uicontrol('units','norm','pos',[.02 .02 .10 .05], ...
'string','help','back','white','callback','helpwin pivotgolf')
bvals = [];
while sum(bvals) < 2
if get(stop,'val'), break, end
pause(.05)
bvals = cell2mat(get(b,'val'));
end
if get(stop,'val')
close(gcf)
return
end
for k = find(bvals)'
switch k
case 1, course = 'magic';
case 2, course = 'testmats';
case {3,4,5,6}, course = k-2;
otherwise, pivotstrat = get(b(k),'string');
end
end
elseif nargin == 1
pivotstrat = 'pick';
end
if isnumeric(course)
rand('state',course)
else
course = lower(course);
end
% Loop over nine holes (matrices).
score = 0;
h = 1;
A = [];
while h <= 9
if isempty(A)
% Generate the matrix.
switch course
case 'magic'
n = 2+ceil(2/3*h);
switch h
case {3,6,9}
A = golub(n);
otherwise
A = magic(n);
end
case 'testmats'
n = h;
switch h
case 1
n = 7;
e = ones(n,1);
A = full(spdiags([e (-3:3)' e],[-1 0 1],n,n));
case 2
n = 5;
A = vander((-2:2)');
case 3
A = gallery(n);
case 4
A = hadamard(n);
case 5
A = gallery(n);
case 6
A = pascal(n);
case 7
n = 6;
A = pascal(n);
A(n,n) = A(n,n)-1;
case 8
n = 5;
A = 27720*hilb(n);
case 9
n = 6;
U = eye(n,n) - triu(ones(n,n),1);
A = U'*U;
end
otherwise
n = ceil(2+3*rand);
A = round(10*(2*rand(n,n)-1));
end
end
% Use LUGUI to compute the LU decomposition
[L,U,p,q] = lugui(A,pivotstrat);
pause(2);
% Score
R = (L*U - A(p,q));
Leps = L.*(abs(L)<1000*norm(L,1)*eps);
Ueps = U.*(abs(U)<1000*norm(U,1)*eps);
show(abs(R)+abs(Leps)+abs(Ueps));
if all(isfinite(R(:)))
s = ceil(4*(norm(R(:),1)+norm(Leps(:),1)+norm(Ueps(:),1))/eps)/4;
else
s = Inf;
end
% Report the score and decide what to do next.
set(gcf,'name','Pivot Pickin'' Golf')
text('units','pixels','pos',[20+50*n,20+(n+2)*30], ...
'fontweight','bold','fontsize',12, ...
'color',[0 0.65 0],'string',sprintf('hole #%d',h))
text('units','pixels','pos',[50*(n-1)+25,55], ...
'fontweight','bold','fontsize',12,'color',[0 0 0.90], ...
'string',sprintf('score = %s, total = %s',sph(s),sph(score+s)))
stop = uicontrol('style','toggle','string','X','fontweight','bold', ...
'back','w','pos',[100*n+75 30*n+65 25 25]);
if isequal(pivotstrat,'pick')
next = uicontrol('units','pixels','pos',[50*(n-1) 10 90 20], ...
'style','toggle','fontweight','bold', ...
'background','white','string','next');
if h==9, set(next,'string','finish'), end
mulligan = uicontrol('units','pixels','pos',[50*(n+1) 10 90 20], ...
'style','toggle','fontweight','bold', ...
'background','white','string','mulligan');
uics = [stop next mulligan];
while all(cell2mat(get(uics,'val'))==0)
drawnow
end
if get(mulligan,'val'), continue, end
else
pause(3)
end
score = score + s;
if get(stop,'val'), break, end
h = h + 1;
A = [];
end
% Final screen
clf
set(gcf,'double','on','name','Pivot Pickin'' Golf', ...
'pos','default','menu','none','numbertitle','off','color','white')
axes('pos',[0 0 1 1]);
axis off
text('units','norm','pos',[.15,.60], ...
'fontweight','bold','fontsize',16,'color',[0 0.65 0], ...
'string',sprintf('Thanks for playing Pivot Pickin'' Golf.'))
if score == 0
text('units','norm','pos',[.15,.50],...
'fontweight','bold','fontsize',16,'color',[0 0.65 0], ...
'string','Perfect score. Congratulations!')
elseif score == Inf
text('units','norm','pos',[.15,.50], ...
'fontweight','bold','fontsize',16,'color',[0 0.65 0], ...
'color',[0 0.65 0],'string','Your score was infinite.')
text('units','norm','pos',[.15,.40], ...
'fontweight','bold','fontsize',16,'color',[0 0.65 0], ...
'color',[0 0.65 0],'string','You have to avoid dividing by zero.')
else
text('units','norm','pos',[.15,.50], ...
'fontweight','bold','fontsize',16,'color',[0 0.65 0], ...
'string',sprintf('Your score was %s eps.',sph(score)))
if score > 100
text('units','norm','pos',[.15,.40], ...
'fontweight','bold','fontsize',16,'color',[0 0.65 0], ...
'string','Better luck next time.')
end
end
uicontrol('style','toggle','units','norm','pos',[.96 .96 .04 .04], ...
'string','X','fontweight','bold','callback','close(gcf)')
if nargout > 0
scoreout = score;
pause(3)
end
%------------------------------------------------------------
function show(A)
% Same code as LUGUI.
clf
axes('pos',[0 0 1 1]);
axis off
[m,n] = size(A);
dx = 100;
dy = 30;
Acolor = [0 0 0];
for j = 1:n
for i = 1:m
t(i,j) = text('units','pixels','string',spf(A(i,j)), ...
'fontname','courier','fontweight','bold','fontsize',14, ...
'horiz','right','color',Acolor, ...
'pos',[20+j*dx 20+(m+2-i)*dy]);
end
end
%------------------------------------------------------------
function s = spf(aij)
% Subfunction to format text strings
if aij == 0
f = '%10.0f';
elseif (abs(aij) < 1.e-4) | (abs(aij) >= 1.e4)
f = '%10.1e';
else
f = '%10.4f';
end
s = sprintf(f,aij);
%------------------------------------------------------------
function s = sph(x)
% Format text strings that are integer multiples of 1/4.
if x == 0
f = '%d';
elseif x == round(x);
f = '%1.0f';
elseif x == round(2*x)/2;
f = '%2.1f';
else
f = '%3.2f';
end
s = sprintf(f,x);