Mercurial > hg > octave-nkf
changeset 12608:59e2460acae1
make piecewise polynomial (pp) functions more compatible
author | Kai Habel <kai.habel@gmx.de> |
---|---|
date | Wed, 23 Feb 2011 08:11:40 +0100 |
parents | 3047363c376d |
children | f3ba948328e9 |
files | scripts/ChangeLog scripts/general/interp1.m scripts/polynomial/mkpp.m scripts/polynomial/pchip.m scripts/polynomial/ppder.m scripts/polynomial/ppint.m scripts/polynomial/ppjumps.m scripts/polynomial/ppval.m scripts/polynomial/spline.m scripts/polynomial/unmkpp.m |
diffstat | 10 files changed, 352 insertions(+), 269 deletions(-) [+] |
line wrap: on
line diff
--- a/scripts/ChangeLog +++ b/scripts/ChangeLog @@ -1,3 +1,12 @@ +2011-04-15 Kai Habel <kai.habel@gmx.de> + + * general/interp1.m, polynomial/mkpp.m, polynomial/pchip.m, + polynomial/ppder.m, polynomial/ppint.m, polynomial/ppjumps.m, + polynomial/ppval.m, polynomial/spline.m, polynomial/unmkpp.m: + Make functions more compatible with respect to handling of + picewise polynoms (pp). Rename pp-struct elements. + Handle nD-arguments correctly. Tests added. + 2011-04-13 Rik <octave@nomad.inbox5.com> * help/__makeinfo__.m: Simplify function by using regular expressions. @@ -414,11 +423,6 @@ * statistics/base/mean.m: Also accept logical values. -2011-02-10 Carlo de Falco <kingcrimson@tiscali.it> - - * linear-algebra/gmres.m: New file implementing the GMRES - iterative method for solving linear systems. - 2011-02-08 Ben Abbott <bpabbott@mac.com> * plot/__go_draw_axes__.m: Properly set fontspec for legends.
--- a/scripts/general/interp1.m +++ b/scripts/general/interp1.m @@ -43,7 +43,7 @@ ## Piecewise cubic Hermite interpolating polynomial ## ## @item 'cubic' -## Cubic interpolation from four nearest neighbors +## Cubic interpolation (same as @code{pchip}) ## ## @item 'spline' ## Cubic spline interpolation---smooth first and second derivatives @@ -112,7 +112,7 @@ method = "linear"; extrap = NA; xi = []; - pp = false; + ispp = false; firstnumeric = true; if (nargin > 2) @@ -123,7 +123,7 @@ if (strcmp ("extrap", arg)) extrap = "extrap"; elseif (strcmp ("pp", arg)) - pp = true; + ispp = true; else method = arg; endif @@ -138,7 +138,7 @@ endfor endif - if (isempty (xi) && firstnumeric && ! pp) + if (isempty (xi) && firstnumeric && ! ispp) xi = y; y = x; x = 1:numel(y); @@ -150,9 +150,8 @@ szx = size (xi); if (isvector (y)) y = y(:); - elseif (isvector (xi)) - szx = length (xi); endif + szy = size (y); y = y(:,:); [ny, nc] = size (y); @@ -191,147 +190,85 @@ switch (method) case "nearest" - if (pp) - yi = mkpp ([x(1); (x(1:nx-1)+x(2:nx))/2; x(nx)], y, szy(2:end)); + pp = mkpp ([x(1); (x(1:nx-1)+x(2:nx))/2; x(nx)], shiftdim (y, 1), szy(2:end)); + pp.orient = "first"; + + if (ispp) + yi = pp; else - idx = lookup (0.5*(x(1:nx-1)+x(2:nx)), xi) + 1; - yi = y(idx,:); + yi = ppval (pp, reshape (xi, szx)); endif case "*nearest" - if (pp) - yi = mkpp ([x(1); x(1)+[0.5:(nx-1)]'*dx; x(nx)], y, szy(2:end)); + pp = mkpp ([x(1), x(1)+[0.5:(nx-1)]*dx, x(nx)], shiftdim (y, 1), szy(2:end)); + pp.orient = "first"; + if (ispp) + yi = pp; else - idx = max (1, min (ny, floor((xi-x(1))/dx+1.5))); - yi = y(idx,:); + yi = ppval(pp, reshape (xi, szx)); endif case "linear" dy = diff (y); - dx = diff (x); - if (pp) - coefs = [dy./dx, y(1:nx-1)]; - xx = x; - if (have_jumps) - ## Omit zero-size intervals. - coefs(jumps) = []; - xx(jumps) = []; - endif - yi = mkpp (xx, coefs, szy(2:end)); + dx = diff (x); + dx = repmat (dx, [1 size(dy)(2:end)]); + coefs = [(dy./dx).'(:), y(1:nx-1, :).'(:)]; + xx = x; + + if (have_jumps) + ## Omit zero-size intervals. + coefs(jumps, :) = []; + xx(jumps) = []; + endif + + pp = mkpp (xx, coefs, szy(2:end)); + pp.orient = "first"; + + if (ispp) + yi = pp; else - ## find the interval containing the test point - idx = lookup (x, xi, "lr"); - ## use the endpoints of the interval to define a line - s = (xi - x(idx))./dx(idx); - yi = bsxfun (@times, s, dy(idx,:)) + y(idx,:); - if (have_jumps) - ## Fix the corner cases of discontinuities at boundaries. - ## Internal discontinuities already handled correctly. - if (jumps (1)) - mask = xi < x(1); - yi(mask,:) = y(1*ones (1, sum (mask)),:); - endif - if (jumps(nx-1)) - mask = xi >= x(nx); - yi(mask,:) = y(nx*ones (1, sum (mask)),:); - endif - endif + yi = ppval(pp, reshape (xi, szx)); endif + case "*linear" dy = diff (y); - if (pp) - yi = mkpp (x(1) + [0:ny-1]*dx, [dy./dx, y(1:end-1)], szy(2:end)); + coefs = [(dy/dx).'(:), y(1:nx-1, :).'(:)]; + pp = mkpp (x, coefs, szy(2:end)); + pp.orient = "first"; + + if (ispp) + yi = pp; else - ## find the interval containing the test point - t = (xi - x(1))/dx + 1; - idx = max (1, min (ny - 1, floor (t))); + yi = ppval(pp, reshape (xi, szx)); + endif - ## use the endpoints of the interval to define a line - s = t - idx; - yi = bsxfun (@times, s, dy(idx,:)) + y(idx,:); - endif - case {"pchip", "*pchip"} + case {"pchip", "*pchip", "cubic", "*cubic"} if (nx == 2 || starmethod) x = linspace (x(1), x(nx), ny); endif - ## Note that pchip's arguments are transposed relative to interp1 - if (pp) - yi = pchip (x.', y.'); - yi.d = szy(2:end); - else - yi = pchip (x.', y.', xi.').'; - endif - - case {"cubic", "*cubic"} - if (nx < 4 || ny < 4) - error ("interp1: table too short"); - endif - - ## FIXME Is there a better way to treat pp return and *cubic - if (starmethod && ! pp) - ## From: Miloje Makivic - ## http://www.npac.syr.edu/projects/nasa/MILOJE/final/node36.html - t = (xi - x(1))/dx + 1; - idx = max (min (floor (t), ny-2), 2); - t = t - idx; - t2 = t.*t; - tp = 1 - 0.5*t; - a = (1 - t2).*tp; - b = (t2 + t).*tp; - c = (t2 - t).*tp/3; - d = (t2 - 1).*t/6; - J = ones (1, nc); - - yi = a(:,J) .* y(idx,:) + b(:,J) .* y(idx+1,:) ... - + c(:,J) .* y(idx-1,:) + d(:,J) .* y(idx+2,:); + + if (ispp) + y = shiftdim (reshape (y, szy), 1); + yi = pchip (x, y); else - if (starmethod) - x = linspace (x(1), x(nx), ny).'; - nx = ny; - endif - - idx = lookup (x(2:nx-1), xi, "lr"); - - ## Construct cubic equations for each interval using divided - ## differences (computation of c and d don't use divided differences - ## but instead solve 2 equations for 2 unknowns). Perhaps - ## reformulating this as a lagrange polynomial would be more efficient. - i = 1:nx-3; - J = ones (1, nc); - dx = diff (x); - dx2 = x(i+1).^2 - x(i).^2; - dx3 = x(i+1).^3 - x(i).^3; - a = diff (y, 3)./dx(i,J).^3/6; - b = (diff (y(1:nx-1,:), 2)./dx(i,J).^2 - 6*a.*x(i+1,J))/2; - c = (diff (y(1:nx-2,:), 1) - a.*dx3(:,J) - b.*dx2(:,J))./dx(i,J); - d = y(i,:) - ((a.*x(i,J) + b).*x(i,J) + c).*x(i,J); - - if (pp) - xs = [x(1);x(3:nx-2)]; - yi = mkpp ([x(1);x(3:nx-2);x(nx)], - [a(:), (b(:) + 3.*xs(:,J).*a(:)), ... - (c(:) + 2.*xs(:,J).*b(:) + 3.*xs(:,J)(:).^2.*a(:)), ... - (d(:) + xs(:,J).*c(:) + xs(:,J).^2.*b(:) + ... - xs(:,J).^3.*a(:))], szy(2:end)); - else - yi = ((a(idx,:).*xi(:,J) + b(idx,:)).*xi(:,J) ... - + c(idx,:)).*xi(:,J) + d(idx,:); - endif + y = shiftdim (y, 1); + yi = pchip (x, y, reshape (xi, szx)); endif case {"spline", "*spline"} if (nx == 2 || starmethod) x = linspace(x(1), x(nx), ny); endif - ## Note that spline's arguments are transposed relative to interp1 - if (pp) - yi = spline (x.', y.'); - yi.d = szy(2:end); + + if (ispp) + y = shiftdim (reshape (y, szy), 1); + yi = spline (x, y); else - yi = spline (x.', y.', xi.').'; + y = shiftdim (y, 1); + yi = spline (x, y, reshape (xi, szx)); endif otherwise error ("interp1: invalid method '%s'", method); endswitch - if (! pp) + if (! ispp) if (! ischar (extrap)) ## determine which values are out of range and set them to extrap, ## unless extrap == "extrap". @@ -339,10 +276,24 @@ maxx = max (x(1), x(nx)); outliers = xi < minx | ! (xi <= maxx); # this catches even NaNs - yi(outliers, :) = extrap; + if (size_equal (outliers, yi)) + yi(outliers) = extrap; + yi = reshape (yi, szx); + elseif (!isvector (yi)) + if (strcmp (method, "pchip") || strcmp (method, "*pchip") + ||strcmp (method, "cubic") || strcmp (method, "*cubic") + ||strcmp (method, "spline") || strcmp (method, "*spline")) + yi(:, outliers) = extrap; + yi = shiftdim(yi, 1); + else + yi(outliers, :) = extrap; + endif + else + yi(outliers.') = extrap; + endif endif - - yi = reshape (yi, [szx, szy(2:end)]); + else + yi.orient = "first"; endif endfunction @@ -394,6 +345,7 @@ %! %-------------------------------------------------------- %! % confirm that interpolated function matches the original +##FIXME: add test for n-d arguments here ## For each type of interpolated test, confirm that the interpolated ## value at the knots match the values at the knots. Points away @@ -595,7 +547,6 @@ %!assert (interp1(1:2,1:2,1.4,"nearest"),1); %!error interp1(1,1,1, "linear"); %!assert (interp1(1:2,1:2,1.4,"linear"),1.4); -%!error interp1(1:3,1:3,1, "cubic"); %!assert (interp1(1:4,1:4,1.4,"cubic"),1.4); %!assert (interp1(1:2,1:2,1.1, "spline"), 1.1); %!assert (interp1(1:3,1:3,1.4,"spline"),1.4); @@ -604,7 +555,6 @@ %!assert (interp1(1:2:4,1:2:4,1.4,"*nearest"),1); %!error interp1(1,1,1, "*linear"); %!assert (interp1(1:2:4,1:2:4,[0,1,1.4,3,4],"*linear"),[NA,1,1.4,3,NA]); -%!error interp1(1:3,1:3,1, "*cubic"); %!assert (interp1(1:2:8,1:2:8,1.4,"*cubic"),1.4); %!assert (interp1(1:2,1:2,1.3, "*spline"), 1.3); %!assert (interp1(1:2:6,1:2:6,1.4,"*spline"),1.4); @@ -612,5 +562,5 @@ %!assert (interp1([3,2,1],[3,2,2],2.5),2.5) %!assert (interp1 ([1,2,2,3,4],[0,1,4,2,1],[-1,1.5,2,2.5,3.5], "linear", "extrap"), [-2,0.5,4,3,1.5]) -%!assert (interp1 ([4,4,3,2,0],[0,1,4,2,1],[1.5,4,4.5], "linear"), [0,1,NA]) +%!assert (interp1 ([4,4,3,2,0],[0,1,4,2,1],[1.5,4,4.5], "linear"), [1.75,1,NA]) %!assert (interp1 (0:4, 2.5), 1.5)
--- a/scripts/polynomial/mkpp.m +++ b/scripts/polynomial/mkpp.m @@ -17,50 +17,66 @@ ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- -## @deftypefn {Function File} {@var{pp} =} mkpp (@var{x}, @var{p}) -## @deftypefnx {Function File} {@var{pp} =} mkpp (@var{x}, @var{p}, @var{d}) +## @deftypefn {Function File} {@var{pp} =} mkpp (@var{breaks}, @var{coefs}) +## @deftypefnx {Function File} {@var{pp} =} mkpp (@var{breaks}, @var{coefs}, @var{d}) +## +## Construct a piece-wise polynomial (pp) structure from sample points +## @var{breaks} and coefficients @var{coefs}. @var{breaks} must be a vector of +## strictly increasing values. The number of intervals is given by +## @code{@var{ni} = length (@var{breaks}) - 1}. +## When @var{m} is the polynomial order @var{coefs} must be of +## size: @var{ni} x @var{m} + 1. ## -## Construct a piecewise polynomial structure from sample points -## @var{x} and coefficients @var{p}. The i-th row of @var{p}, -## @code{@var{p} (@var{i},:)}, contains the coefficients for the polynomial -## over the @var{i}-th interval, ordered from highest to -## lowest. There must be one row for each interval in @var{x}, so -## @code{rows (@var{p}) == length (@var{x}) - 1}. +## The i-th row of @var{coefs}, +## @code{@var{coefs} (@var{i},:)}, contains the coefficients for the polynomial +## over the @var{i}-th interval, ordered from highest (@var{m}) to +## lowest (@var{0}). ## -## @var{p} may also be a multi-dimensional array, specifying a vector-valued -## or array-valued polynomial. The shape is determined by @var{d}. If @var{d} -## is -## not given, the default is @code{size (p)(1:end-2)}. If @var{d} is given, the -## leading dimensions of @var{p} are reshaped to conform to @var{d}. +## @var{coefs} may also be a multi-dimensional array, specifying a vector-valued +## or array-valued polynomial. In that case the polynomial order is defined +## by the length of the last dimension of @var{coefs}. +## The size of first dimension(s) are given by the scalar or +## vector @var{d}. If @var{d} is not given it is set to @code{1}. +## In any case @var{coefs} is reshaped to a 2d matrix of +## size @code{[@var{ni}*prod(@var{d} @var{m})] } ## ## @seealso{unmkpp, ppval, spline} ## @end deftypefn function pp = mkpp (x, P, d) + + # check number of arguments if (nargin < 2 || nargin > 3) print_usage (); endif - pp.x = x(:); - n = length (x) - 1; - if (n < 1) + + # check x + if (length (x) < 2) error ("mkpp: at least one interval is needed"); endif - nd = ndims (P); - k = size (P, nd); - if (nargin < 3) - if (nd == 2) - d = 1; - else - d = prod (size (P)(1:nd-1)); - endif + + if (!isvector (x)) + error ("mkpp: x must be a vector"); endif - pp.d = d; - pp.P = P = reshape (P, prod (d), [], k); - pp.orient = 0; + + len = length (x) - 1; + dP = length (size (P)); - if (size (P, 2) != n) - error ("mkpp: num intervals in X doesn't match num polynomials in P"); - endif + pp = struct ("form", "pp", + "breaks", x(:).', + "coefs", [], + "pieces", len, + "order", prod (size (P)) / len, + "dim", 1); + + if (nargin == 3) + pp.dim = d; + pp.order /= prod (d); + endif + + dim_vec = [pp.pieces * prod(pp.dim), pp.order]; + pp.coefs = reshape (P, dim_vec); + endfunction %!demo # linear interpolation @@ -72,3 +88,25 @@ %! xi=linspace(0,pi,50); %! plot(x,t,"x",xi,ppval(pp,xi)); %! legend("control","interp"); + +%!shared b,c,pp +%! b = 1:3; c = 1:24; pp=mkpp(b,c); +%!assert (pp.pieces,2); +%!assert (pp.order,12); +%!assert (pp.dim,1); +%!assert (size(pp.coefs),[2,12]); +%! pp=mkpp(b,c,2); +%!assert (pp.pieces,2); +%!assert (pp.order,6); +%!assert (pp.dim,2); +%!assert (size(pp.coefs),[4,6]); +%! pp=mkpp(b,c,3); +%!assert (pp.pieces,2); +%!assert (pp.order,4); +%!assert (pp.dim,3); +%!assert (size(pp.coefs),[6,4]); +%! pp=mkpp(b,c,[2,3]); +%!assert (pp.pieces,2); +%!assert (pp.order,2); +%!assert (pp.dim,[2,3]); +%!assert (size(pp.coefs),[12,2]);
--- a/scripts/polynomial/pchip.m +++ b/scripts/polynomial/pchip.m @@ -27,8 +27,8 @@ ## ## The variable @var{x} must be a strictly monotonic vector (either ## increasing or decreasing). While @var{y} can be either a vector or -## array. In the case where @var{y} is a vector, it must have a length -## of @var{n}. If @var{y} is an array, then the size of @var{y} must +## an array. In the case where @var{y} is a vector, it must have the +## length @var{n}. If @var{y} is an array, then the size of @var{y} must ## have the form ## @tex ## $$[s_1, s_2, \cdots, s_k, n]$$ @@ -73,15 +73,22 @@ print_usage (); endif + ## make row vector x = x(:).'; n = length (x); ## Check the size and shape of y if (isvector (y)) - y = y(:).'; + y = y(:).'; ##row vector szy = size (y); + if !(size_equal (x, y)) + error ("pchip: length of X and Y must match") + endif else szy = size (y); + if (n != szy(end)) + error ("pchip: length of X and last dimension of Y must match") + endif y = reshape (y, [prod(szy(1:end-1)), szy(end)]); endif @@ -94,16 +101,12 @@ error("pchip: X must be strictly monotonic"); endif - if (columns (y) != n) - error("pchip: size of X and Y must match"); - endif - - f1 = y(:,1:n-1); + f1 = y(:, 1:n-1); ## Compute derivatives. d = __pchip_deriv__ (x, y, 2); - d1 = d(:,1:n-1); - d2 = d(:,2:n); + d1 = d(:, 1:n-1); + d2 = d(:, 2:n); ## This is taken from SLATEC. h = diag (h); @@ -114,14 +117,12 @@ c3 = del1 + del2; c2 = -c3 - del1; c3 = c3 / h; - coeffs = cat (3, c3, c2, d1, f1); - pp = mkpp (x, coeffs, szy(1:end-1)); - if (nargin == 2) - ret = pp; - else - ret = ppval (pp, xi); + ret = mkpp (x, coeffs, szy(1:end-1)); + + if (nargin == 3) + ret = ppval (ret, xi); endif endfunction @@ -138,7 +139,7 @@ %! %------------------------------------------------------------------- %! % confirm that pchip agreed better to discontinuous data than spline -%!shared x,y +%!shared x,y,y2,pp,yi1,yi2,yi3 %! x = 0:8; %! y = [1, 1, 1, 1, 0.5, 0, 0, 0, 0]; %!assert (pchip(x,y,x), y); @@ -148,3 +149,23 @@ %!assert (isempty(pchip(x',y',[]))); %!assert (isempty(pchip(x,y,[]))); %!assert (pchip(x,[y;y],x), [pchip(x,y,x);pchip(x,y,x)]) +%!assert (pchip(x,[y;y],x'), [pchip(x,y,x);pchip(x,y,x)]) +%!assert (pchip(x',[y;y],x), [pchip(x,y,x);pchip(x,y,x)]) +%!assert (pchip(x',[y;y],x'), [pchip(x,y,x);pchip(x,y,x)]) +%!test +%! x=(0:8)*pi/4;y=[sin(x);cos(x)]; +%! y2(:,:,1)=y;y2(:,:,2)=y+1;y2(:,:,3)=y-1; +%! pp=pchip(x,shiftdim(y2,2)); +%! yi1=ppval(pp,(1:4)*pi/4); +%! yi2=ppval(pp,repmat((1:4)*pi/4,[5,1])); +%! yi3=ppval(pp,[pi/2,pi]); +%!assert(size(pp.coefs),[48,4]); +%!assert(pp.pieces,8); +%!assert(pp.order,4); +%!assert(pp.dim,[3,2]); +%!assert(ppval(pp,pi),[0,-1;1,0;-1,-2],1e-14); +%!assert(yi3(:,:,2),ppval(pp,pi),1e-14); +%!assert(yi3(:,:,1),[1,0;2,1;0,-1],1e-14); +%!assert(squeeze(yi1(1,2,:)),[1/sqrt(2); 0; -1/sqrt(2);-1],1e-14); +%!assert(size(yi2),[3,2,5,4]); +%!assert(squeeze(yi2(1,2,3,:)),[1/sqrt(2); 0; -1/sqrt(2);-1],1e-14); \ No newline at end of file
--- a/scripts/polynomial/ppder.m +++ b/scripts/polynomial/ppder.m @@ -17,28 +17,54 @@ ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- -## @deftypefn {Function File} {@var{ppd} =} ppder (@var{pp}) -## Compute the piecewise derivative of the piecewise polynomial struct @var{pp}. +## @deftypefn {Function File} {ppd =} ppder (pp, m) +## Computes the piecewise @var{m}-th derivative of a piecewise polynomial +## struct @var{pp}. If @var{m} is omitted the first derivate is +## calculated. ## @seealso{mkpp, ppval, ppint} ## @end deftypefn -function ppd = ppder (pp) - if (nargin != 1) +function ppd = ppder (pp, m) + + if ((nargin < 1) || (nargin > 2)) print_usage (); + elseif (nargin == 1) + m = 1; endif - if (! isstruct (pp)) + + if !(isstruct (pp) && strcmp (pp.form, "pp")) error ("ppder: PP must be a structure"); endif [x, p, n, k, d] = unmkpp (pp); - p = reshape (p, [], k); - if (k <= 1) - pd = zeros (rows (p), 1); - k = 1; + + if (k - m <= 0) + x = [x(1) x(end)]; + pd = zeros (prod (d), 1); else - k -= 1; - pd = p(:,1:k) * diag (k:-1:1); + f = k : -1 : 1; + ff = bincoeff (f, m + 1) .* factorial (m + 1) ./ f; + k -= m; + pd = p(:,1:k) * diag (ff(1:k)); endif + ppd = mkpp (x, pd, d); endfunction +%!shared x,y,pp,ppd +%! x=0:8;y=[x.^2;x.^3+1];pp=spline(x,y); +%! ppd=ppder(pp); +%!assert(ppval(ppd,x),[2*x;3*x.^2],1e-14) +%!assert(ppd.order,3) +%! ppd=ppder(pp,2); +%!assert(ppval(ppd,x),[2*ones(size(x));6*x],1e-14) +%!assert(ppd.order,2) +%! ppd=ppder(pp,3); +%!assert(ppd.order,1) +%!assert(ppd.pieces,8) +%!assert(size(ppd.coefs),[16,1]) +%! ppd=ppder(pp,4); +%!assert(ppd.order,1) +%!assert(ppd.pieces,1) +%!assert(size(ppd.coefs),[2,1]) +%!assert(ppval(ppd,x),zeros(size(y)),1e-14)
--- a/scripts/polynomial/ppint.m +++ b/scripts/polynomial/ppint.m @@ -28,7 +28,7 @@ if (nargin < 1 || nargin > 2) print_usage (); endif - if (! isstruct (pp)) + if (! isstruct (pp) && strcmp (pp.form, "pp")) error ("ppint: PP must be a structure"); endif @@ -39,17 +39,20 @@ pi = p / diag (k:-1:1); k += 1; if (nargin == 1) - pi(:,k) = 0; + pi(:, k) = 0; else - pi(:,k) = repmat (c(:), n, 1); + pi(:, k) = repmat (c(:), n, 1); endif ppi = mkpp (x, pi, d); - ## Adjust constants so the the result is continuous. - - jumps = reshape (ppjumps (ppi), prod (d), n-1); - ppi.P(:,2:n,k) -= cumsum (jumps, 2); - + tmp = -cumsum (ppjumps (ppi), length (d) + 1); + ppi.coefs(prod(d)+1:end, k) = tmp(:); + endfunction +%!shared x,y,pp,ppi +%! x=0:8;y=[ones(size(x));x+1];pp=spline(x,y); +%! ppi=ppint(pp); +%!assert(ppval(ppi,x),[x;0.5*x.^2+x],1e-14) +%!assert(ppi.order,5)
--- a/scripts/polynomial/ppjumps.m +++ b/scripts/polynomial/ppjumps.m @@ -28,29 +28,31 @@ if (nargin != 1) print_usage (); endif - if (! isstruct (pp)) + + if (! isstruct (pp) && strcmp (pp.form, "pp")) error ("ppjumps: PP must be a structure"); endif ## Extract info. - x = pp.x; - P = pp.P; - d = pp.d; - [nd, n, k] = size (P); + [x, P, n, k, d] = unmkpp(pp); + nd = length (d) + 1; ## Offsets. - dx = diff (x(1:n)).'; - dx = dx(ones (1, nd), :); # spread (do nothing in 1D) + dx = diff(x(1:n)); + dx = repmat (dx, [prod(d), 1]); + dx = reshape (dx, [d, n-1]); + dx = shiftdim (dx, nd - 1); - ## Use Horner scheme to get limits from the left. - llim = P(:,1:n-1,1); - for i = 2:k; + ## Use Horner scheme. + if (k>1) + llim = shiftdim (reshape (P(1:(n-1) * prod(d), 1), [d, n-1]), nd - 1); + endif + + for i = 2 : k; llim .*= dx; - llim += P(:,1:n-1,i); + llim += shiftdim (reshape (P(1:(n-1) * prod (d), i), [d, n-1]), nd - 1); endfor - - rlim = P(:,2:n,k); # limits from the right - jumps = reshape (rlim - llim, [d, n-1]); - + + rlim = shiftdim (ppval (pp, x(2:end-1)), nd - 1); + jumps = shiftdim (rlim - llim, 1); endfunction -
--- a/scripts/polynomial/ppval.m +++ b/scripts/polynomial/ppval.m @@ -18,16 +18,18 @@ ## -*- texinfo -*- ## @deftypefn {Function File} {@var{yi} =} ppval (@var{pp}, @var{xi}) -## Evaluate piecewise polynomial @var{pp} at the points @var{xi}. -## If @var{pp} is scalar-valued, the result is an array of the same shape as -## @var{xi}. -## Otherwise, the size of the result is @code{[pp.d, length(@var{xi})]} if -## @var{xi} is a vector, or @code{[pp.d, size(@var{xi})]} if it is a -## multi-dimensional array. If pp.orient is 1, the dimensions are permuted as +## Evaluate piece-wise polynomial structure @var{pp} at the points @var{xi}. +## If @var{pp} describes a scalar polynomial function, the result is an +## array of the same shape as @var{xi}. +## Otherwise, the size of the result is @code{[pp.dim, length(@var{xi})]} if +## @var{xi} is a vector, or @code{[pp.dim, size(@var{xi})]} if it is a +## multi-dimensional array. +## +##, the dimensions are permuted as ## in interp1, to ## @code{[pp.d, length(@var{xi})]} and @code{[pp.d, size(@var{xi})]} ## respectively. -## @seealso{mkpp, unmkpp, spline} +## @seealso{mkpp, unmkpp, spline, pchip, interp1} ## @end deftypefn function yi = ppval (pp, xi) @@ -35,48 +37,85 @@ if (nargin != 2) print_usage (); endif - if (! isstruct (pp)) - error ("ppval: PP must be a structure"); + if (! isstruct (pp) && strcmp (pp.form, "pp")) + error ("ppval: expects a pp-form structure"); endif ## Extract info. - x = pp.x; - P = pp.P; - d = pp.d; - k = size (P, 3); - nd = size (P, 1); - - ## Determine resulting shape. - if (d == 1) # scalar case - yisz = size (xi); - elseif (isvector (xi)) # this is special - yisz = [d, length(xi)]; - else # general - yisz = [d, size(xi)]; + [x, P, n, k, d] = unmkpp (pp); + + ## dimension checks + sxi = size (xi); + if (isvector (xi)) + xi = xi(:).'; endif + + nd = length (d); ## Determine intervals. - xi = xi(:); xn = numel (xi); - idx = lookup (x, xi, "lr"); + P = reshape (P, [d, n * k]); + P = shiftdim (P, nd); + P = reshape (P, [n, k, d]); + Pidx = P(idx(:), :);#2d matrix size x: coefs*prod(d) y: prod(sxi) + + if (isvector(xi)) + Pidx = reshape (Pidx, [xn, k, d]); + Pidx = shiftdim (Pidx, 1); + dimvec = [d, xn]; + else + Pidx = reshape (Pidx, [sxi, k, d]); + Pidx = shiftdim (Pidx, length (sxi)); + dimvec = [d, sxi]; + end + ndv = length (dimvec); + ## Offsets. - dx = (xi - x(idx)).'; - dx = dx(ones (1, nd), :); # spread (do nothing in 1D) + dx = (xi - x(idx)); + dx = repmat (dx, [prod(d), 1]); + dx = reshape (dx, dimvec); + dx = shiftdim (dx, ndv - 1); ## Use Horner scheme. - yi = P(:,idx,1); - for i = 2:k; + yi = Pidx; + if (k > 1) + yi = shiftdim (reshape (Pidx(1,:), dimvec), ndv - 1); + endif + + for i = 2 : k; yi .*= dx; - yi += P(:,idx,i); + yi += shiftdim (reshape (Pidx(i,:), dimvec), ndv - 1); endfor - + ## Adjust shape. - yi = reshape (yi, yisz); - if (d != 1 && pp.orient == 1) - ## Switch dimensions to match interp1 order. - yi = shiftdim (yi, length (d)); + if ((numel (xi) > 1) || (length (d) == 1)) + yi = reshape (shiftdim (yi, 1), dimvec); endif + if (isvector (xi) && (d == 1)) + yi = reshape (yi, sxi); + elseif (isfield (pp, "orient") && strcmp (pp.orient, "first")) + yi = shiftdim(yi, nd); + endif + + ## + #if (d == 1) + # yi = reshape (yi, sxi); + #endif + endfunction + +%!shared b,c,pp,pp2,xi,abserr +%! b = 1:3; c = ones(2); pp=mkpp(b,c);abserr = 1e-14;pp2=mkpp(b,[c;c],2); +%! xi = [1.1 1.3 1.9 2.1]; +%!assert (ppval(pp,1.1), 1.1, abserr); +%!assert (ppval(pp,2.1), 1.1, abserr); +%!assert (ppval(pp,xi), [1.1 1.3 1.9 1.1], abserr); +%!assert (ppval(pp,xi.'), [1.1 1.3 1.9 1.1].', abserr); +%!assert (ppval(pp2,1.1), [1.1;1.1], abserr); +%!assert (ppval(pp2,2.1), [1.1;1.1], abserr); +%!assert (ppval(pp2,xi), [1.1 1.3 1.9 1.1;1.1 1.3 1.9 1.1], abserr); +%!assert (ppval(pp2,xi'), [1.1 1.3 1.9 1.1;1.1 1.3 1.9 1.1], abserr); +%!assert (size(ppval(pp2,[xi;xi])), [2 2 4]);
--- a/scripts/polynomial/spline.m +++ b/scripts/polynomial/spline.m @@ -83,15 +83,15 @@ ## Check the size and shape of y ndy = ndims (y); szy = size (y); - if (ndy == 2 && (szy(1) == 1 || szy(2) == 1)) - if (szy(1) == 1) + if (ndy == 2 && (szy(1) == n || szy(2) == n)) + if (szy(2) == n) a = y.'; else a = y; szy = fliplr (szy); endif else - a = reshape (y, [prod(szy(1:end-1)), szy(end)]).'; + a = shiftdim (reshape (y, [prod(szy(1:end-1)), szy(end)]), 1); endif for k = (1:columns (a))(any (isnan (a))) @@ -120,9 +120,9 @@ if (n == 2) d = (dfs + dfe) / (x(2) - x(1)) ^ 2 + ... - 2 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 3; + 2 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 3; c = (-2 * dfs - dfe) / (x(2) - x(1)) - ... - 3 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 2; + 3 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 2; b = dfs; a = a(1,:); @@ -132,7 +132,7 @@ a = a(1:n-1,:); else if (n == 3) - dg = 1.5 * h(1) - 0.5 * h(2); + dg = 1.5 * h(1) - 0.5 * h(2); c(2:n-1,:) = 1/dg(1); else dg = 2 * (h(1:n-2) .+ h(2:n-1)); @@ -153,9 +153,9 @@ endif c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * dfs - - c(2,:) * h(1)) / (2 * h(1)); + - c(2,:) * h(1)) / (2 * h(1)); c(n,:) = - (3 / h(n-1) * (a(n,:) - a(n-1,:)) - 3 * dfe - + c(n-1,:) * h(n-1)) / (2 * h(n-1)); + + c(n-1,:) * h(n-1)) / (2 * h(n-1)); b(1:n-1,:) = diff (a) ./ h(1:n-1, idx) ... - h(1:n-1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); d = diff (c) ./ (3 * h(1:n-1, idx)); @@ -229,15 +229,14 @@ - h(1:n-1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); d = diff (c) ./ (3 * h(1:n-1, idx)); - d = d(1:n-1,:); - c = c(1:n-1,:); - b = b(1:n-1,:); - a = a(1:n-1,:); + d = d(1:n-1,:);d = d.'(:); + c = c(1:n-1,:);c = c.'(:); + b = b(1:n-1,:);b = b.'(:); + a = a(1:n-1,:);a = a.'(:); endif endif - coeffs = cat (3, d.', c.', b.', a.'); - ret = mkpp (x, coeffs, szy(1:end-1)); + ret = mkpp (x, cat (2, d, c, b, a), szy(1:end-1)); if (nargin == 3) ret = ppval (ret, xi); @@ -263,6 +262,9 @@ %!assert (isempty(spline(x',y',[]))); %!assert (isempty(spline(x,y,[]))); %!assert (spline(x,[y;y],x), [spline(x,y,x);spline(x,y,x)],abserr) +%!assert (spline(x,[y;y],x'), [spline(x,y,x);spline(x,y,x)],abserr) +%!assert (spline(x',[y;y],x), [spline(x,y,x);spline(x,y,x)],abserr) +%!assert (spline(x',[y;y],x'), [spline(x,y,x);spline(x,y,x)],abserr) %! y = cos(x) + i*sin(x); %!assert (spline(x,y,x), y, abserr) %!assert (real(spline(x,y,x)), real(y), abserr);
--- a/scripts/polynomial/unmkpp.m +++ b/scripts/polynomial/unmkpp.m @@ -50,15 +50,13 @@ if (nargin == 0) print_usage (); endif - if (! isstruct (pp)) + if (! (isstruct (pp) && strcmp (pp.form, "pp"))) error ("unmkpp: expecting piecewise polynomial structure"); endif - x = pp.x; - P = pp.P; - n = size (P, 2); - k = size (P, 3); - d = pp.d; - if (d == 1) - P = reshape (P, n, k); - endif + x = pp.breaks; + P = pp.coefs; + n = pp.pieces; + k = pp.order; + d = pp.dim; + endfunction