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);