octave-general 2.0.0-1 / usr / share / octave / packages / general-2.0.0 / doc-cache

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adresamp2


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 -- Function File: [XS, YS] = adresamp2 (X, Y, N, EPS)
     Perform an adaptive resampling of a planar curve.  The arrays X and
     Y specify x and y coordinates of the points of the curve.  On
     return, the same curve is approximated by XS, YS that have length N
     and the angles between successive segments are approximately equal.


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Perform an adaptive resampling of a planar curve.



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cauchy


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 -- Function File: cauchy (N, R, X, F )
     Return the Taylor coefficients and numerical differentiation of a
     function F for the first N-1 coefficients or derivatives using the
     fft.  N is the number of points to evaluate, R is the radius of
     convergence, needs to be chosen less then the smallest singularity,
     X is point to evaluate the Taylor expansion or differentiation.
     For example,

     If X is a scalar, the function F is evaluated in a row vector of
     length N.  If X is a column vector, F is evaluated in a matrix of
     length(x)-by-N elements and must return a matrix of the same size.

          d = cauchy(16, 1.5, 0, @(x) exp(x));
          => d(2) = 1.0000 # first (2-1) derivative of function f (index starts from zero)


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Return the Taylor coefficients and numerical differentiation of a
function F for



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majle


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MAJLE	(Weak) Majorization check
    S = MAJLE(X,Y) checks if the real part of X is (weakly) majorized by
    the real part of Y, where X and Y must be numeric (full or sparse)
    arrays. It returns S=0, if there is no weak majorization of X by Y,
    S=1, if there is a weak majorization of X by Y, or S=2, if there is a
    strong majorization of X by Y. The shapes of X and Y are ignored.
    NUMEL(X) and NUMEL(Y) may be different, in which case one of them is
    appended with zeros to match the sizes with the other and, in case of
    any negative components, a special warning is issued.  

    S = MAJLE(X,Y,MAJLETOL) allows in addition to specify the tolerance in
    all inequalities [S,Z] = MAJLE(X,Y,MAJLETOL) also outputs a row vector
    Z, which appears in the definition of the (weak) majorization. In the
    traditional case, where the real vectors X and Y are of the same size,
    Z = CUMSUM(SORT(Y,'descend')-SORT(X,'descend')). Here, X is weakly
    majorized by Y, if MIN(Z)>0, and strongly majorized if MIN(Z)=0, see
    http://en.wikipedia.org/wiki/Majorization

    The value of MAJLETOL depends on how X and Y have been computed, i.e.,
    on what the level of error in X or Y is. A good minimal starting point
    should be MAJLETOL=eps*MAX(NUMEL(X),NUMEL(Y)). The default is 0. 

    % Examples:
    x = [2 2 2]; y = [1 2 3]; s = majle(x,y)
    % returns the value 2.
    x = [2 2 2]; y = [1 2 4]; s = majle(x,y)
    % returns the value 1.
    x = [2 2 2]; y = [1 2 2]; s = majle(x,y)
    % returns the value 0.
    x = [2 2 2]; y = [1 2 2]; [s,z] = majle(x,y)
    % also returns the vector z = [ 0 0 -1].
    x = [2 2 2]; y = [1 2 2]; s = majle(x,y,1)
    % returns the value 2.
    x = [2 2]; y = [1 2 2]; s = majle(x,y)
    % returns the value 1 and warns on tailing with zeros
    x = [2 2]; y = [-1 2 2]; s = majle(x,y)
    % returns the value 0 and gives two warnings on tailing with zeros
    x = [2 -inf]; y = [4 inf]; [s,z] = majle(x,y)
    % returns s = 1 and z = [Inf   Inf].
    x = [2 inf]; y = [4 inf]; [s,z] = majle(x,y)
    % returns  s = 1 and z = [NaN NaN] and a warning on NaNs in z.
    x=speye(2); y=sparse([0 2; -1 1]); s = majle(x,y) 
    % returns the value 2.
    x = [2 2; 2 2]; y = [1 3 4]; [s,z] = majle(x,y) %and 
    x = [2 2; 2 2]+i; y = [1 3 4]-2*i; [s,z] = majle(x,y)
    % both return s = 2 and z = [2 3 2 0]. 
    x = [1 1 1 1 0]; y = [1 1 1 1 1 0 0]'; s = majle(x,y)
    % returns the value 1 and warns on tailing with zeros

    % One can use this function to check numerically the validity of the
    Schur-Horn,Lidskii-Mirsky-Wielandt, and Gelfand-Naimark theorems: 
    clear all; n=100; majleTol=n*n*eps;
    A = randn(n,n); A = A'+A; eA = -sort(-eig(A)); dA = diag(A);
    majle(dA,eA,majleTol) % returns the value 2
    % which is the Schur-Horn theorem; and 
    B=randn(n,n); B=B'+B; eB=-sort(-eig(B)); 
    eAmB=-sort(-eig(A-B));
    majle(eA-eB,eAmB,majleTol) % returns the value 2 
    % which is the Lidskii-Mirsky-Wielandt theorem; finally
    A = randn(n,n); sA = -sort(-svd(A)); 
    B = randn(n,n); sB = -sort(-svd(B));
    sAB = -sort(-svd(A*B));
    majle(log2(sAB)-log2(sA), log2(sB), majleTol) % retuns the value 2
    majle(log2(sAB)-log2(sB), log2(sA), majleTol) % retuns the value 2
    % which are the log versions of the Gelfand-Naimark theorems



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MAJLE	(Weak) Majorization check
    S = MAJLE(X,Y) checks if the real part of X



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safeprod


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 -- Function File: P = safeprod (X, DIM)
 -- Function File: [P, E] = safeprod (X, DIM)
     This function forms product(s) of elements of the array X along the
     dimension specified by DIM, analogically to 'prod', but avoids
     overflows and underflows if possible.  If called with 2 output
     arguments, P and E are computed so that the product is 'P * 2^E'.

     See also: prod,log2.


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This function forms product(s) of elements of the array X along the
dimension sp



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tablify


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 -- Function File: [Y1, ...] = tablify (X1, ...)

     Create a table out of the input arguments, if possible.  The table
     is created by extending row and column vectors to like dimensions.
     If the dimensions of input vectors are not commensurate an error
     will occur.  Dimensions are commensurate if they have the same
     number of rows and columns, a single row and the same number of
     columns, or the same number of rows and a single column.  In other
     words, vectors will only be extended along singleton dimensions.

     For example:

          [a, b] = tablify ([1 2; 3 4], 5)
          => a = [ 1, 2; 3, 4 ]
          => b = [ 5, 5; 5, 5 ]
          [a, b, c] = tablify (1, [1 2 3 4], [5;6;7])
          =>
          b = [ 1 1 1 1; 1 1 1 1; 1 1 1 1]
          => b = [ 1 2 3 4; 1 2 3 4; 1 2 3 4]
          => c = [ 5 5 5 5; 6 6 6 6; 7 7 7 7 ]

     The following example attempts to expand vectors that do not have
     commensurate dimensions and will produce an error.

          tablify([1 2],[3 4 5])

     Note that use of array operations and broadcasting is more
     efficient for many situations.

     See also: common_size.


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Create a table out of the input arguments, if possible.



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unresamp2


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 -- Function File: [XS, YS] = unresamp2 (X, Y, N)
     Perform a uniform resampling of a planar curve.  The arrays X and Y
     specify x and y coordinates of the points of the curve.  On return,
     the same curve is approximated by XS, YS that have length N and the
     distances between successive points are approximately equal.


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Perform a uniform resampling of a planar curve.



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unvech


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 -- Function File: M = unvech (V, SCALE)
     Performs the reverse of 'vech' on the vector V.

     Given a Nx1 array V describing the lower triangular part of a
     matrix (as obtained from 'vech'), it returns the full matrix.

     The upper triangular part of the matrix will be multiplied by SCALE
     such that 1 and -1 can be used for symmetric and antisymmetric
     matrix respectively.  SCALE must be a scalar and defaults to 1.

     See also: vech, ind2sub.


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Performs the reverse of 'vech' on the vector V.



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ztvals


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 -- Function File: function ztvals (X, TOL)
     Replaces tiny elements of the vector X by zeros.  Equivalent to
            X(abs(X) < TOL * norm (X, Inf)) = 0
     TOL specifies the chopping tolerance.  It defaults to 1e-10 for
     double precision and 1e-5 for single precision inputs.


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Replaces tiny elements of the vector X by zeros.