function varargout = interp1(varargin)
%INTERP1 1-D interpolation (table lookup)
% YI = INTERP1(X,Y,XI) interpolates to find YI, the values of the
% underlying function Y at the points in the array XI. X must be a
% vector of length N.
% If Y is a vector, then it must also have length N, and YI is the
% same size as XI. If Y is an array of size [N,D1,D2,...,Dk], then
% the interpolation is performed for -Dk value
% in Y(i,:,:,...,:).
% If XI is a vector of length M, then YI has size [M,D1,D2,...,Dk].
% If XI is an array of size [M1,M2,...,Mj], then YI is of size
% [M1,M2,...,Mj,D1,D2,...,Dk].
%
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% YI = INTERP1(Y,XI) assumes X = 1:N, where N is LENGTH(Y)
% for vector Y or SIZE(Y,1) for array Y.
%
书法培训那里好啊% Interpolation is the same operation as "table lookup". Described in
% "table lookup" terms, the "table" is [X,Y] and INTERP1 "looks-up"
% the elements of XI in X, and, bad upon their location, returns
% values YI interpolated within the elements of Y.
%
% YI = INTERP1(X,Y,XI,METHOD) specifies alternate methods.
% The default is linear interpolation. U an empty matrix [] to specify
% the default. Available methods are:
%
% 'nearest' - nearest neighbor interpolation
% 'linear' - linear interpolation
% 'spline' - piecewi cubic spline interpolation (SPLINE)
% 'pchip' - shape-prerving piecewi cubic interpolation
% 'cubic' - same as 'pchip'
% 'v5cubic' - the cubic interpolation from MATLAB 5, which does not
% extrapolate and us 'spline' if X is not equally
% spaced.
%
% YI = INTERP1(X,Y,XI,METHOD,'extrap') us the specified method for
% extrapolation for any elements of XI outside the interval spanned by X.
% Alternatively, YI = INTERP1(X,Y,XI,METHOD,EXTRAPVAL) replaces
% the values outside of the interval spanned by X with EXTRAPVAL.
% NaN and 0 are often ud for EXTRAPVAL. The default extrapolation
% behavior with four input arguments is 'extrap' for 'spline' and 'pchip'
% and EXTRAPVAL = NaN for the other methods.
%
% PP = INTERP1(X,Y,METHOD,'pp') will u the specified method to
% generate the ppform (piecewi polynomial form) of Y. The method may be
% any of the above except for 'v5cubic'. PP may then be evaluated via脱下
% PPVAL. PPVAL(PP,XI) is the same as INTERP1(X,Y,XI,METHOD,'extrap').
%
% For example, generate a coar sine curve and interpolate over a
% finer abscissa:
% x = 0:10; y = sin(x); xi = 0:.25:10;
% yi = interp1(x,y,xi); plot(x,y,'o',xi,yi)
%
% For a multi-dimensional example, we construct a table of functional
% values:
% x = [1:10]'; y = [ x.^2, x.^3, x.^4 ];
% xi = [ 1.5, 1.75; 7.5, 7.75]; yi = interp1(x,y,xi);
%
% creates 2-by-2 matrices of interpolated function values, one matrix for
% each of the 3 functions. yi will be of size 2-by-2-by-3.
%
% Class support for inputs X, Y,
XI, EXTRAPVAL:
% float: double, single
%
% See also INTERP1Q, INTERPFT, SPLINE, PCHIP, INTERP2, INTERP3, INTERPN, PPVAL.
% Copyright 1984-2009 The MathWorks, Inc.
% $Revision: 5.41.4.12 $ $Date: 2009/04/21 03:25:36 $
%
% Determine input arguments.
% The following syntaxes do not have X as the first input:
% INTERP1(Y,XI)
% INTERP1(Y,XI,METHOD)
% INTERP1(Y,XI,METHOD,'extrap')
% INTERP1(Y,XI,METHOD,EXTRAPVAL)
xOfft = 1;
if (nargin==2) || ...
(nargin==3 && ischar(varargin{3})) || ...
(nargin==4 && ~(ischar(varargin{4}) || impty(varargin{4})) || ...
(nargin==4 && strcmp(varargin{4}, 'extrap')));
xOfft = 0;
end
ppOutput = fal;
% PP = INTERP1(X,Y,METHOD,'pp')
if nargin>=4 && ischar(varargin{3}) && iqual('pp',varargin{4})
ppOutput = true;
if (nargin > 4)
error('MATLAB:interp1:ppOutput', ...
'U 4 inputs for PP=INTERP1(X,Y,METHOD,''pp'').')
end
end
% Process Y in INTERP1(Y,...) and INTERP1(X,Y,...)
y = varargin{1+xOfft};
siz_y = size(y);
% y may be an ND array, but collap it down to a 2D yMat. If yMat is
% a vector, it is a column vector.
if isvector(y)
if size(y,1) == 1
% Prefer column vectors for y
yMat = y.';
n = siz_y(2);
el
yMat = y;
n = siz_y(1);
end
ds = 1;
prodDs = 1;
el
n = siz_y(1);
ds = siz_y(2:end);
prodDs = prod(ds);
yMat = reshape(y,[n prodDs]);
end
% Process X in INTERP1(X,Y,...), or supply default for INTERP1(Y,...)
if xOfft
x = varargin{xOfft};
if ~isvector(x)
error('MATLAB:interp1:Xvector','X must be a vector.');
end
if length(x) ~= n
if isvector(y)
error('MATLAB:interp1:YInvalidNumRows', ...
'X and Y must be of the same length.')
el
error('MATLAB:interp1:YInvalidNumRows', ...
'LENGTH(X) and SIZE(Y,1) must be the same.');
end
end
% Prefer column vectors for x
xCol = x(:);
el
xCol = (1:n)';
end
% Process XI in INTERP1(Y,XI,...) and INTERP1(X,Y,XI,...)
% Avoid syntax PP = INTERP1(X,Y,METHOD,'pp')
if ~ppOutput
xi = varargin{2+xOfft};
siz_xi = size(xi);
% xi may be an ND array, but flatten it to a column vector xiCol
xiCol = xi(:);
% The size of the output YI
if isvector(y)
% Y is a vector so size(YI) == size(XI)
siz_yi = siz_xi;
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el
if isvector(xi)
% Y is not a vector but XI is
siz_yi = [length(xi) ds];
el
% Both Y and XI are non-vectors
siz_yi = [siz_xi ds];
end
end
end
if xOfft && ~isreal(x)
error('MATLAB:interp1:ComplexX','X should be a real vector.')
end
if ~ppOutput && ~isreal(xi)
error('MATLAB:interp1:ComplexInterpPts', ...
'The interpolation points XI should be real.')
end
% Error check for NaN values
in X and Y
% check for NaN's
if xOfft && (any(isnan(xCol)))
error('MATLAB:interp1:NaNinX','NaN is not an appropriate value for X.');
end
% NANS are allowed as a value for F(X), since a function may be undefined
% for a given value.
if any(isnan(yMat(:)))
warning('MATLAB:interp1:NaNinY', ...
['NaN found in Y, interpolation at undefined values \n\t',...
' will result in undefined values.']);
end
if (n < 2)
if ppOutput || ~impty(xi)
error('MATLAB:interp1:NotEnoughPts', ...
'There should be at least two data points.')
el
yi = zeros(siz_yi,superiorfloat(x,y,xi));
varargout{1} = yi;
return
end
end
% Process METHOD in
% PP = INTERP1(X,Y,METHOD,'pp')
% YI = INTERP1(Y,XI,METHOD,...)
% YI = INTERP1(X,Y,XI,METHOD,...)
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% including explicit specification of the default by an empty input.
if ppOutput
if impty(varargin{3})
method = 'linear';
el
method = varargin{3};
end
el
if nargin >= 3+xOfft && ~impty(varargin{3+xOfft})
method = varargin{3+xOfft};
el
method = 'linear';
end
end
% The v5 option, '*method', asrts that x is equally spaced.
eqsp = (method(1) == '*');
if eqsp
method(1) = [];
end
% INTERP1([X,]Y,XI,METHOD,'extrap') and INTERP1([X,]Y,Xi,METHOD,EXTRAPVAL)
if ~ppOutput
if nargin >= 4+xOfft
extrapval = varargin{4+xOfft};
el
switch method(1)
ca {'s','p','c'}
extrapval = 'extrap';
otherwi
extrapval = NaN;
end
end
end
% Start the algorithm
% We now have column vector xCol, column vector or 2D matrix yMat and
% column vector xiCol.
if xOfftassure
if ~eqsp
h = diff(xCol);
eqsp = (norm(diff(h),Inf) <= eps(norm(xCol,Inf)));
if any(~isfinite(xCol))
eqsp = 0; % if an INF in x, x is not equally spaced
end
end
if eqsp
h = (xCol(n)-xCol(1))/(n-1);
end
el
h = 1;
eqsp = 1;
end
if any(h < 0)
[xCol,p] = sort(xCol);
yMat = yMat(p,:);
if eqsp
h = -h;
el
h = diff(xCol);
end
end
if any(h == 0)
error('MATLAB:interp1:RepeatedValuesX', ...
'The values of X should be distinct.');
end
% PP = INTERP1(X,Y,METHOD,'pp')
if nargin==4 && ischar(varargin{3}) && iqual('pp',varargin{4})
% obtain pp form of output
pp = ppinterp;
varargout{1} = pp;
return
end
% Interpolate
numelXi = length(xiCol);
p = [];
switch method(1)
ca 's' % 'spline'
% spline is oriented opposite to interp1
yiMat = spline(xCol.',yMat.',xiCol.').';
ca {'c','p'} % 'cubic' or 'pchip'
% pchip is oriented opposite to interp1
yiMat = pchip(xCol.',yMat.',xiCol.').';
otherwi % 'nearest', 'linear', 'v5cubic'
yiMat = zeros(numelXi,prodDs
,superiorfloat(xCol,yMat,xiCol));
if ~eqsp && any(diff(xiCol) < 0)
[xiCol,p] = sort(xiCol);
el
p = 1:numelXi;
end
% Find indices of subintervals, x(k) <= u < x(k+1),
% or u < x(1) or u >= x(m-1).
if impty(xiCol)
k = xiCol;
elif eqsp
k = min(max(1+floor((xiCol-xCol(1))/h),1),n-1);
el
[~,k] = histc(xiCol,xCol);
k(xiCol<xCol(1) | ~isfinite(xiCol)) = 1;
k(xiCol>=xCol(n)) = n-1;
end
switch method(1)
ca 'n' % 'nearest'
i = find(xiCol >= (xCol(k)+xCol(k+1))/2);
k(i) = k(i)+1;
yiMat(p,:) = yMat(k,:);
ca 'l' % 'linear'
if eqsp
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s = (xiCol - xCol(k))/h;
el
s = (xiCol - xCol(k))./h(k);
end
for j = 1:prodDs
yiMat(p,j) = yMat(k,j) + s.*(yMat(k+1,j)-yMat(k,j));
end
ca 'v' % 'v5cubic'
extrapval = NaN;
if eqsp
% Data are equally spaced
s = (xiCol - xCol(k))/h;
s2 = s.*s;
s3 = s.*s2;
% Add extra points for first and last interval
yMat = [3*yMat(1,:)-3*yMat(2,:)+yMat(3,:); ...
yMat; ...
3*yMat(n,:)-3*yMat(n-1,:)+yMat(n-2,:)];
for j = 1:prodDs
yiMat(p,j) = (yMat(k,j).*(-s3+2*s2-s) + ...
yMat(k+1,j).*(3*s3-5*s2+2) + ...
yMat(k+2,j).*(-3*s3+4*s2+s) + ...
yMat(k+3,j).*(s3-s2))/2;
end
el
% Data are not equally spaced
% spline is oriented opposite to interp1
yiMat = spline(xCol.',yMat.',xiCol.').';
end
otherwi
error('MATLAB:interp1:InvalidMethod','Invalid method.')
end
end
% Override extrapolation
if ~iqual(extrapval,'extrap')
if ischar(extrapval)
error('MATLAB:interp1:InvalidExtrap', 'Invalid extrap option.')
football的音标elif ~isscalar(extrapval)
error('MATLAB:interp1:NonScalarExtrapValue',...
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'EXTRAP option must be a scalar.')
end
if impty(p)
p = 1 : numelXi;
end
outOfBounds = xiCol<xCol(1) | xiCol>xCol(n);
yiMat(p(outOfBounds),:) = extrapval;
end
% Reshape result, possibly to an ND array
yi = reshape(yiMat,siz_yi);
varargout{1} = yi;
%-------------------------------------------------------------------------%
function pp = ppinterp
%PPINTERP ppform interpretation.
switch method(1)
ca 'n' % nearest
breaks = [xCol(1); ...
(xCol(1:end-1)+xCol(2:end))/2; ...
xCol(end)].';
coefs = yMat.';
pp = mkpp(breaks,coefs,ds);
ca 'l' % linear
breaks = xCol.';
page1 = (diff(yMat)./repmat(diff(xCol),[1, prodDs])).';
page2 = (reshape(yMat(1:end-1,:),[n-1, prodDs])).';
coefs = cat(3,page1,page2);
pp = mkpp(breaks,coefs,ds);
ca {'p', 'c'} % pchip and cubic
pp = pchip(xCol.',reshape(yMat.',[ds, n]));
ca 's' % spline
pp = spline(xCol.',reshape(yMat.',[ds, n]));
ca 'v' % v5cubic
b = diff(xCol);
if norm(diff(b),Inf) <= eps(norm(xCol,Inf))
% data are equally spaced
a = repmat(b,[1 prodDs]).';
yReorg = [3*yMat(1,:)-3*yMat(2,:)+yMat(3,:); ...
yMat; ...
3*yMat(n,:)-3*yMat(n-1,:)+yMat(n-2,:)];
y1 = yReorg(1:end-3,:).';
y2 = yReorg(2:end-2,:).';
y3 = yReorg(3:end-1,:).';
y4 = yReorg(4:end,:).';
breaks = xCol.';
page1 = (-y1+3*y2-3*y3+y4)./(2*a.^3);
page2 = (2*y1-5*y2+4*y3-y4)./(2*a.^2);
page3 = (-y1+y3)./(2*a);
page4 = y2;
coefs = cat(3,page1,page2,page3,page4);
pp = mkpp(breaks,coefs,ds);
el
% data are not equally spaced
pp = spline(xCol.',reshape(yMat.',[ds, n]));
end
otherwi
error('MATLAB:interp1:ppinterp:UnknownMethod', ...
'Unrecognized method.');
end
% Even if method is 'spline' or 'pchip', we still need to record that the
% input data Y was oriented according to INTERP1's rules.
% Thus PPVAL will return YI oriented according to INTERP1's rules and
hangen% YI = INTERP1(X,Y,XI,METHOD) will be the same as
% YI = PPVAL(INTERP1(X,Y,METHOD,'pp'),XI)
end % PPINTERP
end % INTERP1
