t=[0,0,0,0,1,2,3,3,3,4,5,6,6,6,6]

t =

  Columns 1 through 9

     0     0     0     0     1     2     3     3     3

  Columns 10 through 15

     4     5     6     6     6     6

xx=linspace(0,6,101);
plot(xx,BSplineEval(1,4,t,xx))
plot(xx,BSplineEval(2,4,t,xx))
plot(xx,BSplineEval(3,4,t,xx))
plot(xx,BSplineEval(4,4,t,xx))
plot(xx,BSplineEval(,4,t,xx))
??? plot(xx,BSplineEval(,4,t,xx))
                        |
Error: Expression or statement is incorrect--possibly
unbalanced (, {, or [.

plot(xx,BSplineEval(5,4,t,xx))
plot(xx,BSplineEval(6,4,t,xx))
plot(xx,BSplineEval(7,4,t,xx))
plot(xx,BSplineEval(8,4,t,xx))
xx=linspace(-1,1,101);
plot(xx,abs(xx).^(1/2))
xx=linspace(0,6,101);
plot(xx,BSplineEval(6,4,t,xx))
uiopen('/home/accounts/personale/clrmrc90/aa1213/advanced_numerical_analysis/orderbspline.m', true);
% Modify expression to add input arguments.
% Example:
%   a = [1 2 3; 4 5 6]; 
%   foo(a);

orderbspline

x =

   -1.0000
   -1.0000
   -0.7500
   -0.2500
         0
         0
    0.2500
    0.7500
    1.0000
    1.0000

t

t =

   -1.0000
   -1.0000
   -1.0000
   -1.0000
   -0.5000
         0
         0
         0
    0.5000
    1.0000
    1.0000
    1.0000
    1.0000

x

x =

   -1.0000
   -1.0000
   -0.7500
   -0.2500
         0
    0.2500
    0.7500
    1.0000
    1.0000

length(x)

ans =

     9

length(t)

ans =

    13

plot(t,zeros(size(t)),'x',x,zeros(size(x)),'o')
help spapi
 SPAPI Spline interpolation.
 
    SPAPI(KNOTS,X,Y)      (with both KNOTS and X vectors)
    returns the spline  f  (if any) of order
          k := length(KNOTS) - length(X)
    with knot sequence KNOTS for which Y(:,j) equals f(X(j)) , all j.
    This is taken in the osculatory sense in case some data sites are
    repeated, i.e., in the sense that  D^m(j) f(X(j)) = Y(:,j)  with
    m(j) := #{ i<j : X(i) = X(j) }, and  D^m(j) f  the m(j)-th derivative of f.
    If you don't want this interpretation, call SPAPI with an additional
    (fourth) input argument in which case the average of all data values
    at the same data site is matched at that site.
    The data values, Y(:,j), may be scalars, vectors, matrices, or even
    ND-arrays.
 
    SPAPI(K,X,Y)          (with K a positive integer)
    is the same as  SPAPI(APTKNT(sort(X),K),X,Y), i.e., it uses APTKNT to
    obtain from K and X a suitable knot sequence.
 
    SPAPI({KNOTS1,...,KNOTSm},{X1,...,Xm},Y)
    returns in SP the m-variate tensor-product spline of coordinate order
       ki := length(KNOTSi) - length(Xi)
    with knot sequence KNOTSi in the i-th variable, i=1,...,m, for which
       Y(:,j1,...,jm) = f(X1(j1),...,Xm(jm)),  all j := (j1,...,jm) .
    As in the univariate case, KNOTSi may also be a positive integer, in
    which case the i-th knot sequence is obtained from Xi via APTKNT.
    Note the possibility of interpolating to d-valued data. However,
    in contrast to the univariate case, if the data to be interpolated are
    scalar-valued, then the input array Y is permitted to be m-dimensional,
    in which case
    Y(j1,...,jm) = f(X1(j1),...,Xm(jm)),  all j := (j1,...,jm) .
    Multiplicities in the sequences Xi lead to osculatory interpolation just
    as in the univariate case.
 
    For example, if the points in the vector  t  are all distinct, then
 
       sp = spapi(augknt(t,4,2),[t t],[cos(t) -sin(t)]);
 
    provides the C^1 piecewise cubic Hermite interpolant to the function
    f(x) = cos(x)  at the points in  t . If matching of slopes is only
    required at some subsequence  s  of  t  but that includes the leftmost
    and the rightmost point in  t , one would use instead
 
       sp = spapi( augknt([t s],4), [t s], [cos(t) -sin(s)] );
 
    or even just
 
       sp = spapi( 4, [t s], [cos(t) -sin(s)] );
 
    and the last works even if s fails to include the extreme points of t,
    and produces even a C^2 piecewise cubic interpolant to these Hermite data.
 
    As another example,
 
       sp = spapi( {[0 0 1 1],[0 0 1 1]}, {[0 1],[0 1]}, [0 0;0 1] );
 
    constructs the bilinear interpolant to values at the corners of a square,
    as would the statement
 
       sp = spapi({2,2},{[0 1],[0 1]},[0 0;0 1]);
 
    As another example, the statements
 
       x = -2:.5:2; y=-1:.25:1; [xx, yy] = ndgrid(x,y);
       z = exp(-(xx.^2+yy.^2));
       sp = spapi({3,4},{x,y},z);
       fnplt(sp)
 
    produce the picture of an interpolant (piecewise quadratic in x,
    piecewise cubic in y) to a bivariate function.
    Use of MESHGRID instead of NDGRID here would produce an error.
 
    As an illustration of osculatory interpolation to gridded data, here
    is complete bicubic interpolation, with the data explicitly derived
    from the bicubic polynomial  g(x,y) = x^3y^3, to make it easy for
    you to see exactly where the slopes and slopes of slopes (i.e., cross
    derivatives) must be placed in the data values supplied. Since our  g  is
    a bicubic polynomial, its interpolant, f , must be  g  itself.
    We test this.
 
       sites = {[0,1],[0,2]}; coefs = zeros(4,4); coefs(1,1) = 1;
       g = ppmak(sites,coefs);
       Dxg = fnval(fnder(g,[1,0]),sites);
       Dyg = fnval(fnder(g,[0,1]),sites);
       Dxyg = fnval(fnder(g,[1,1]),sites);
       f = spapi({4,4}, {sites{1}([1,2,1,2]),sites{2}([1,2,1,2])}, ...
                [fnval(g,sites), Dyg ; ...
                 Dxg.'         , Dxyg]);
       if any( squeeze( fnbrk(fn2fm(f,'pp'), 'c') ) - coefs )
         'something went wrong', end
 
    SPAPI(...,'noderiv') does not do osculatory interpolation but, instead, 
    merely averages all data values at the same site.
 
    See also <a href="matlab:help spaps">spaps</a>, <a href="matlab:help spap2">spap2</a>.

    Reference page in Help browser
       <a href="matlab:doc spapi">doc spapi</a>

x

x =

   -1.0000
   -1.0000
   -0.7500
   -0.2500
         0
    0.2500
    0.7500
    1.0000
    1.0000

help spline
 SPLINE Cubic spline data interpolation.
    PP = SPLINE(X,Y) provides the piecewise polynomial form of the 
    cubic spline interpolant to the data values Y at the data sites X,
    for use with the evaluator PPVAL and the spline utility UNMKPP.
    X must be a vector.
    If Y is a vector, then Y(j) is taken as the value to be matched at X(j), 
    hence Y must be of the same length as X  -- see below for an exception
    to this.
    If Y is a matrix or ND array, then Y(:,...,:,j) is taken as the value to
    be matched at X(j),  hence the last dimension of Y must equal length(X) --
    see below for an exception to this.
 
    YY = SPLINE(X,Y,XX) is the same as  YY = PPVAL(SPLINE(X,Y),XX), thus
    providing, in YY, the values of the interpolant at XX.  For information
    regarding the size of YY see PPVAL.
 
    Ordinarily, the not-a-knot end conditions are used. However, if Y contains
    two more values than X has entries, then the first and last value in Y are
    used as the endslopes for the cubic spline.  If Y is a vector, this
    means:
        f(X) = Y(2:end-1),  Df(min(X))=Y(1),    Df(max(X))=Y(end).
    If Y is a matrix or N-D array with SIZE(Y,N) equal to LENGTH(X)+2, then
    f(X(j)) matches the value Y(:,...,:,j+1) for j=1:LENGTH(X), then
    Df(min(X)) matches Y(:,:,...:,1) and Df(max(X)) matches Y(:,:,...:,end).
 
    Example:
    This generates a sine-like spline curve and samples it over a finer mesh:
        x = 0:10;  y = sin(x);
        xx = 0:.25:10;
        yy = spline(x,y,xx);
        plot(x,y,'o',xx,yy)
 
    Example:
    This illustrates the use of clamped or complete spline interpolation where
    end slopes are prescribed. In this example, zero slopes at the ends of an 
    interpolant to the values of a certain distribution are enforced:
       x = -4:4; y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];
       cs = spline(x,[0 y 0]);
       xx = linspace(-4,4,101);
       plot(x,y,'o',xx,ppval(cs,xx),'-');
 
    Class support for inputs x, y, xx:
       float: double, single
 
    See also <a href="matlab:help interp1">interp1</a>, <a href="matlab:help pchip">pchip</a>, <a href="matlab:help ppval">ppval</a>, <a href="matlab:help unmkpp">unmkpp</a>, <a href="matlab:help mkpp">mkpp</a>, <a href="matlab:help splines">splines</a> (The <a href="matlab:help spline">spline</a> Toolbox).

    Reference page in Help browser
       <a href="matlab:doc spline">doc spline</a>

% Modify expression to add input arguments.
% Example:
%   a = [1 2 3; 4 5 6]; 
%   foo(a);

orderbspline
% Modify expression to add input arguments.
% Example:
%   a = [1 2 3; 4 5 6]; 
%   foo(a);

orderbspline
% Modify expression to add input arguments.
% Example:
%   a = [1 2 3; 4 5 6]; 
%   foo(a);

orderbspline
% Modify expression to add input arguments.
% Example:
%   a = [1 2 3; 4 5 6]; 
%   foo(a);

orderbspline
help spapi
 SPAPI Spline interpolation.
 
    SPAPI(KNOTS,X,Y)      (with both KNOTS and X vectors)
    returns the spline  f  (if any) of order
          k := length(KNOTS) - length(X)
    with knot sequence KNOTS for which Y(:,j) equals f(X(j)) , all j.
    This is taken in the osculatory sense in case some data sites are
    repeated, i.e., in the sense that  D^m(j) f(X(j)) = Y(:,j)  with
    m(j) := #{ i<j : X(i) = X(j) }, and  D^m(j) f  the m(j)-th derivative of f.
    If you don't want this interpretation, call SPAPI with an additional
    (fourth) input argument in which case the average of all data values
    at the same data site is matched at that site.
    The data values, Y(:,j), may be scalars, vectors, matrices, or even
    ND-arrays.
 
    SPAPI(K,X,Y)          (with K a positive integer)
    is the same as  SPAPI(APTKNT(sort(X),K),X,Y), i.e., it uses APTKNT to
    obtain from K and X a suitable knot sequence.
 
    SPAPI({KNOTS1,...,KNOTSm},{X1,...,Xm},Y)
    returns in SP the m-variate tensor-product spline of coordinate order
       ki := length(KNOTSi) - length(Xi)
    with knot sequence KNOTSi in the i-th variable, i=1,...,m, for which
       Y(:,j1,...,jm) = f(X1(j1),...,Xm(jm)),  all j := (j1,...,jm) .
    As in the univariate case, KNOTSi may also be a positive integer, in
    which case the i-th knot sequence is obtained from Xi via APTKNT.
    Note the possibility of interpolating to d-valued data. However,
    in contrast to the univariate case, if the data to be interpolated are
    scalar-valued, then the input array Y is permitted to be m-dimensional,
    in which case
    Y(j1,...,jm) = f(X1(j1),...,Xm(jm)),  all j := (j1,...,jm) .
    Multiplicities in the sequences Xi lead to osculatory interpolation just
    as in the univariate case.
 
    For example, if the points in the vector  t  are all distinct, then
 
       sp = spapi(augknt(t,4,2),[t t],[cos(t) -sin(t)]);
 
    provides the C^1 piecewise cubic Hermite interpolant to the function
    f(x) = cos(x)  at the points in  t . If matching of slopes is only
    required at some subsequence  s  of  t  but that includes the leftmost
    and the rightmost point in  t , one would use instead
 
       sp = spapi( augknt([t s],4), [t s], [cos(t) -sin(s)] );
 
    or even just
 
       sp = spapi( 4, [t s], [cos(t) -sin(s)] );
 
    and the last works even if s fails to include the extreme points of t,
    and produces even a C^2 piecewise cubic interpolant to these Hermite data.
 
    As another example,
 
       sp = spapi( {[0 0 1 1],[0 0 1 1]}, {[0 1],[0 1]}, [0 0;0 1] );
 
    constructs the bilinear interpolant to values at the corners of a square,
    as would the statement
 
       sp = spapi({2,2},{[0 1],[0 1]},[0 0;0 1]);
 
    As another example, the statements
 
       x = -2:.5:2; y=-1:.25:1; [xx, yy] = ndgrid(x,y);
       z = exp(-(xx.^2+yy.^2));
       sp = spapi({3,4},{x,y},z);
       fnplt(sp)
 
    produce the picture of an interpolant (piecewise quadratic in x,
    piecewise cubic in y) to a bivariate function.
    Use of MESHGRID instead of NDGRID here would produce an error.
 
    As an illustration of osculatory interpolation to gridded data, here
    is complete bicubic interpolation, with the data explicitly derived
    from the bicubic polynomial  g(x,y) = x^3y^3, to make it easy for
    you to see exactly where the slopes and slopes of slopes (i.e., cross
    derivatives) must be placed in the data values supplied. Since our  g  is
    a bicubic polynomial, its interpolant, f , must be  g  itself.
    We test this.
 
       sites = {[0,1],[0,2]}; coefs = zeros(4,4); coefs(1,1) = 1;
       g = ppmak(sites,coefs);
       Dxg = fnval(fnder(g,[1,0]),sites);
       Dyg = fnval(fnder(g,[0,1]),sites);
       Dxyg = fnval(fnder(g,[1,1]),sites);
       f = spapi({4,4}, {sites{1}([1,2,1,2]),sites{2}([1,2,1,2])}, ...
                [fnval(g,sites), Dyg ; ...
                 Dxg.'         , Dxyg]);
       if any( squeeze( fnbrk(fn2fm(f,'pp'), 'c') ) - coefs )
         'something went wrong', end
 
    SPAPI(...,'noderiv') does not do osculatory interpolation but, instead, 
    merely averages all data values at the same site.
 
    See also <a href="matlab:help spaps">spaps</a>, <a href="matlab:help spap2">spap2</a>.

    Reference page in Help browser
       <a href="matlab:doc spapi">doc spapi</a>

uiopen('/home/accounts/personale/clrmrc90/aa1213/advanced_numerical_analysis/decasteljau.m', true);
t=linspace(0,1);
b=[1;2]

b =

     1
     2

t=linspace(0,1,5);
t

t =

         0    0.2500    0.5000    0.7500    1.0000

b

b =

     1
     2

repmat(t,2,1)

ans =

         0    0.2500    0.5000    0.7500    1.0000
         0    0.2500    0.5000    0.7500    1.0000

repmat(b,1,5)

ans =

     1     1     1     1     1
     2     2     2     2     2

repmat(t,2,1).*repmat(b,1,5)

ans =

         0    0.2500    0.5000    0.7500    1.0000
         0    0.5000    1.0000    1.5000    2.0000

a{1}=rand(3);
??? Cell contents assignment to a non-cell array
object.

clear all
a{1}=rand(3);
a{2}=rand(3);
a{1}

ans =

    0.9649    0.9572    0.1419
    0.1576    0.4854    0.4218
    0.9706    0.8003    0.9157

a{2}

ans =

    0.7922    0.0357    0.6787
    0.9595    0.8491    0.7577
    0.6557    0.9340    0.7431

b=[1;2]

b =

     1
     2

t
??? Undefined function or variable 't'.

t=linspace(0,1,5);
b=[0,1;1;1/2]
??? Error using ==> vertcat
CAT arguments dimensions are not consistent.

b=[0,1;1,1/2]

b =

         0    1.0000
    1.0000    0.5000

t

t =

         0    0.2500    0.5000    0.7500    1.0000

y = reshape(repmat(b,length(t),1),size(b,1),length(t)*(n+1));
Warning: Could not find an exact (case-sensitive) match
for 'n'.
/home/accounts/personale/clrmrc90/aa1213/advanced_numerical_analysis/N.m
is a case-insensitive match and will be used instead.
You can improve the performance of your code by using
exact
name matches and we therefore recommend that you update
your
usage accordingly. Alternatively, you can disable this
warning using
warning('off','MATLAB:dispatcher:InexactCaseMatch').
This warning will become an error in future releases.
??? Input argument "k" is undefined.

Error in ==> <a href="matlab: opentoline('/home/accounts/personale/clrmrc90/aa1213/advanced_numerical_analysis/N.m',2,0)">N at 2</a>
t = [zeros(1,k-1),linspace(0,k,k+1),k*ones(1,k-1)];

n=1

n =

     1

y = reshape(repmat(b,length(t),1),size(b,1),length(t)*(n+1));
y

y =

  Columns 1 through 5

         0         0         0         0         0
    1.0000    1.0000    1.0000    1.0000    1.0000

  Columns 6 through 10

    1.0000    1.0000    1.0000    1.0000    1.0000
    0.5000    0.5000    0.5000    0.5000    0.5000

t

t =

         0    0.2500    0.5000    0.7500    1.0000

b=[0,1,2;1,1/2,1]

b =

         0    1.0000    2.0000
    1.0000    0.5000    1.0000

n=2

n =

     2

y = reshape(repmat(b,length(t),1),size(b,1),length(t)*(n+1));
y

y =

  Columns 1 through 5

         0         0         0         0         0
    1.0000    1.0000    1.0000    1.0000    1.0000

  Columns 6 through 10

    1.0000    1.0000    1.0000    1.0000    1.0000
    0.5000    0.5000    0.5000    0.5000    0.5000

  Columns 11 through 15

    2.0000    2.0000    2.0000    2.0000    2.0000
    1.0000    1.0000    1.0000    1.0000    1.0000

 2.0000    2.0000    2.0000    2.0000    2.0000
???  2.0000    2.0000    2.0000    2.0000    2.0000
               |
Error: Unexpected MATLAB expression.

mat2cell(y,size(b,1),ones(1,size(b,2))*length(t));
y=mat2cell(y,size(b,1),ones(1,size(b,2))*length(t));
y{1}

ans =

     0     0     0     0     0
     1     1     1     1     1

y{2}

ans =

    1.0000    1.0000    1.0000    1.0000    1.0000
    0.5000    0.5000    0.5000    0.5000    0.5000

y{3}

ans =

     2     2     2     2     2
     1     1     1     1     1