Mercurial > hg > octave-nkf
diff doc/interpreter/matrix.txi @ 3369:f37ca3017116
[project @ 1999-11-21 16:26:02 by jwe]
author | jwe |
---|---|
date | Sun, 21 Nov 1999 16:26:08 +0000 |
parents | bfe1573bd2ae |
children | 0b88d26ed552 |
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--- a/doc/interpreter/matrix.txi +++ b/doc/interpreter/matrix.txi @@ -27,34 +27,9 @@ The @code{find} function is also useful in determining which elements of a matrix meet a specified condition. -@deftypefn {Built-in Function} {} any (@var{x}) -For a vector argument, return 1 if any element of the vector is -nonzero. - -For a matrix argument, return a row vector of ones and -zeros with each element indicating whether any of the elements of the -corresponding column of the matrix are nonzero. For example, +@DOCSTRING(any) -@example -@group -any (eye (2, 4)) - @result{} [ 1, 1, 0, 0 ] -@end group -@end example - -To see if any of the elements of a matrix are nonzero, you can use a -statement like - -@example -any (any (a)) -@end example -@end deftypefn - -@deftypefn {Built-in Function} {} all (@var{x}) -The function @code{all} behaves like the function @code{any}, except -that it returns true only if all the elements of a vector, or all the -elements in a column of a matrix, are nonzero. -@end deftypefn +@DOCSTRING(all) Since the comparison operators (@pxref{Comparison Ops}) return matrices of ones and zeros, it is easy to test a matrix for many things, not just @@ -75,250 +50,32 @@ @code{while} statements) Octave treats the test as if you had typed @code{all (all (condition))}. -@deftypefn {Function File} {[@var{err}, @var{y1}, ...] =} common_size (@var{x1}, ...) -Determine if all input arguments are either scalar or of common -size. If so, @var{err} is zero, and @var{yi} is a matrix of the -common size with all entries equal to @var{xi} if this is a scalar or -@var{xi} otherwise. If the inputs cannot be brought to a common size, -errorcode is 1, and @var{yi} is @var{xi}. For example, - -@example -@group -[errorcode, a, b] = common_size ([1 2; 3 4], 5) - @result{} errorcode = 0 - @result{} a = [ 1, 2; 3, 4 ] - @result{} b = [ 5, 5; 5, 5 ] -@end group -@end example - -@noindent -This is useful for implementing functions where arguments can either -be scalars or of common size. -@end deftypefn +@DOCSTRING(common_size) -@deftypefn {Function File} {} diff (@var{x}, @var{k}) -If @var{x} is a vector of length @var{n}, @code{diff (@var{x})} is the -vector of first differences -@iftex -@tex - $x_2 - x_1, \ldots{}, x_n - x_{n-1}$. -@end tex -@end iftex -@ifinfo - @var{x}(2) - @var{x}(1), @dots{}, @var{x}(n) - @var{x}(n-1). -@end ifinfo - -If @var{x} is a matrix, @code{diff (@var{x})} is the matrix of column -differences. +@DOCSTRING(diff) -The second argument is optional. If supplied, @code{diff (@var{x}, -@var{k})}, where @var{k} is a nonnegative integer, returns the -@var{k}-th differences. -@end deftypefn - -@deftypefn {Mapping Function} {} isinf (@var{x}) -Return 1 for elements of @var{x} that are infinite and zero -otherwise. For example, - -@example -@group -isinf ([13, Inf, NaN]) - @result{} [ 0, 1, 0 ] -@end group -@end example -@end deftypefn - -@deftypefn {Mapping Function} {} isnan (@var{x}) -Return 1 for elements of @var{x} that are NaN values and zero -otherwise. For example, +@DOCSTRING(isinf) -@example -@group -isnan ([13, Inf, NaN]) - @result{} [ 0, 0, 1 ] -@end group -@end example -@end deftypefn - -@deftypefn {Mapping Function} {} finite (@var{x}) -Return 1 for elements of @var{x} that are NaN values and zero -otherwise. For example, - -@example -@group -finite ([13, Inf, NaN]) - @result{} [ 1, 0, 0 ] -@end group -@end example -@end deftypefn - -@deftypefn {Loadable Function} {} find (@var{x}) -Return a vector of indices of nonzero elements of a matrix. To obtain a -single index for each matrix element, Octave pretends that the columns -of a matrix form one long vector (like Fortran arrays are stored). For -example, +@DOCSTRING(isnan) -@example -@group -find (eye (2)) - @result{} [ 1; 4 ] -@end group -@end example - -If two outputs are requested, @code{find} returns the row and column -indices of nonzero elements of a matrix. For example, +@DOCSTRING(finite) -@example -@group -[i, j] = find (2 * eye (2)) - @result{} i = [ 1; 2 ] - @result{} j = [ 1; 2 ] -@end group -@end example - -If three outputs are requested, @code{find} also returns a vector -containing the nonzero values. For example, - -@example -@group -[i, j, v] = find (3 * eye (2)) - @result{} i = [ 1; 2 ] - @result{} j = [ 1; 2 ] - @result{} v = [ 3; 3 ] -@end group -@end example -@end deftypefn +@DOCSTRING(find) @node Rearranging Matrices, Special Utility Matrices, Finding Elements and Checking Conditions, Matrix Manipulation @section Rearranging Matrices -@deftypefn {Function File} {} fliplr (@var{x}) -Return a copy of @var{x} with the order of the columns reversed. For -example, - -@example -@group -fliplr ([1, 2; 3, 4]) - @result{} 2 1 - 4 3 -@end group -@end example -@end deftypefn - -@deftypefn {Function File} {} flipud (@var{x}) -Return a copy of @var{x} with the order of the rows reversed. For -example, - -@example -@group -flipud ([1, 2; 3, 4]) - @result{} 3 4 - 1 2 -@end group -@end example -@end deftypefn +@DOCSTRING(fliplr) -@deftypefn {Function File} {} rot90 (@var{x}, @var{n}) -Return a copy of @var{x} with the elements rotated counterclockwise in -90-degree increments. The second argument is optional, and specifies -how many 90-degree rotations are to be applied (the default value is 1). -Negative values of @var{n} rotate the matrix in a clockwise direction. -For example, - -@example -@group -rot90 ([1, 2; 3, 4], -1) - @result{} 3 1 - 4 2 -@end group -@end example - -@noindent -rotates the given matrix clockwise by 90 degrees. The following are all -equivalent statements: +@DOCSTRING(flipud) -@example -@group -rot90 ([1, 2; 3, 4], -1) -@equiv{} -rot90 ([1, 2; 3, 4], 3) -@equiv{} -rot90 ([1, 2; 3, 4], 7) -@end group -@end example -@end deftypefn - -@deftypefn {Function File} {} reshape (@var{a}, @var{m}, @var{n}) -Return a matrix with @var{m} rows and @var{n} columns whose elements are -taken from the matrix @var{a}. To decide how to order the elements, -Octave pretends that the elements of a matrix are stored in column-major -order (like Fortran arrays are stored). - -For example, +@DOCSTRING(rot90) -@example -@group -reshape ([1, 2, 3, 4], 2, 2) - @result{} 1 3 - 2 4 -@end group -@end example - -If the variable @code{do_fortran_indexing} is nonzero, the -@code{reshape} function is equivalent to - -@example -@group -retval = zeros (m, n); -retval (:) = a; -@end group -@end example - -@noindent -but it is somewhat less cryptic to use @code{reshape} instead of the -colon operator. Note that the total number of elements in the original -matrix must match the total number of elements in the new matrix. -@end deftypefn - -@deftypefn {Function File} {} shift (@var{x}, @var{b}) -If @var{x} is a vector, perform a circular shift of length @var{b} of -the elements of @var{x}. - -If @var{x} is a matrix, do the same for each column of @var{x}. -@end deftypefn +@DOCSTRING(reshape) -@deftypefn {Loadable Function} {[@var{s}, @var{i}] =} sort (@var{x}) -Return a copy of @var{x} with the elements elements arranged in -increasing order. For matrices, @code{sort} orders the elements in each -column. - -For example, - -@example -@group -sort ([1, 2; 2, 3; 3, 1]) - @result{} 1 1 - 2 2 - 3 3 -@end group -@end example +@DOCSTRING(shift) -The @code{sort} function may also be used to produce a matrix -containing the original row indices of the elements in the sorted -matrix. For example, - -@example -@group -[s, i] = sort ([1, 2; 2, 3; 3, 1]) - @result{} s = 1 1 - 2 2 - 3 3 - @result{} i = 1 3 - 2 1 - 3 2 -@end group -@end example -@end deftypefn +@DOCSTRING(sort) Since the @code{sort} function does not allow sort keys to be specified, it can't be used to order the rows of a matrix according to the values @@ -341,162 +98,24 @@ @end group @end example -@deftypefn {Function File} {} tril (@var{a}, @var{k}) -@deftypefnx {Function File} {} triu (@var{a}, @var{k}) -Return a new matrix formed by extracting extract the lower (@code{tril}) -or upper (@code{triu}) triangular part of the matrix @var{a}, and -setting all other elements to zero. The second argument is optional, -and specifies how many diagonals above or below the main diagonal should -also be set to zero. - -The default value of @var{k} is zero, so that @code{triu} and -@code{tril} normally include the main diagonal as part of the result -matrix. - -If the value of @var{k} is negative, additional elements above (for -@code{tril}) or below (for @code{triu}) the main diagonal are also -selected. - -The absolute value of @var{k} must not be greater than the number of -sub- or super-diagonals. - -For example, +@DOCSTRING(tril) -@example -@group -tril (ones (3), -1) - @result{} 0 0 0 - 1 0 0 - 1 1 0 -@end group -@end example - -@noindent -and +@DOCSTRING(vec) -@example -@group -tril (ones (3), 1) - @result{} 1 1 0 - 1 1 1 - 1 1 1 -@end group -@end example -@end deftypefn - -@deftypefn {Function File} {} vec (@var{x}) -Return the vector obtained by stacking the columns of the matrix @var{x} -one above the other. -@end deftypefn - -@deftypefn {Function File} {} vech (@var{x}) -Return the vector obtained by eliminating all supradiagonal elements of -the square matrix @var{x} and stacking the result one column above the -other. -@end deftypefn +@DOCSTRING(vech) @node Special Utility Matrices, Famous Matrices, Rearranging Matrices, Matrix Manipulation @section Special Utility Matrices -@deftypefn {Built-in Function} {} eye (@var{x}) -@deftypefnx {Built-in Function} {} eye (@var{n}, @var{m}) -Return an identity matrix. If invoked with a single scalar argument, -@code{eye} returns a square matrix with the dimension specified. If you -supply two scalar arguments, @code{eye} takes them to be the number of -rows and columns. If given a vector with two elements, @code{eye} uses -the values of the elements as the number of rows and columns, -respectively. For example, - -@example -@group -eye (3) - @result{} 1 0 0 - 0 1 0 - 0 0 1 -@end group -@end example - -The following expressions all produce the same result: +@DOCSTRING(eye) -@example -@group -eye (2) -@equiv{} -eye (2, 2) -@equiv{} -eye (size ([1, 2; 3, 4]) -@end group -@end example - -For compatibility with @sc{Matlab}, calling @code{eye} with no arguments -is equivalent to calling it with an argument of 1. -@end deftypefn - -@deftypefn {Built-in Function} {} ones (@var{x}) -@deftypefnx {Built-in Function} {} ones (@var{n}, @var{m}) -Return a matrix whose elements are all 1. The arguments are handled -the same as the arguments for @code{eye}. - -If you need to create a matrix whose values are all the same, you should -use an expression like - -@example -val_matrix = val * ones (n, m) -@end example -@end deftypefn +@DOCSTRING(ones) -@deftypefn {Built-in Function} {} zeros (@var{x}) -@deftypefnx {Built-in Function} {} zeros (@var{n}, @var{m}) -Return a matrix whose elements are all 0. The arguments are handled -the same as the arguments for @code{eye}. -@end deftypefn - -@deftypefn {Loadable Function} {} rand (@var{x}) -@deftypefnx {Loadable Function} {} rand (@var{n}, @var{m}) -@deftypefnx {Loadable Function} {} rand (@code{"seed"}, @var{x}) -Return a matrix with random elements uniformly distributed on the -interval (0, 1). The arguments are handled the same as the arguments -for @code{eye}. In -addition, you can set the seed for the random number generator using the -form - -@example -rand ("seed", @var{x}) -@end example - -@noindent -where @var{x} is a scalar value. If called as - -@example -rand ("seed") -@end example +@DOCSTRING(zeros) -@noindent -@code{rand} returns the current value of the seed. -@end deftypefn - -@deftypefn {Loadable Function} {} randn (@var{x}) -@deftypefnx {Loadable Function} {} randn (@var{n}, @var{m}) -@deftypefnx {Loadable Function} {} randn (@code{"seed"}, @var{x}) -Return a matrix with normally distributed random elements. The -arguments are handled the same as the arguments for @code{eye}. In -addition, you can set the seed for the random number generator using the -form +@DOCSTRING(rand) -@example -randn ("seed", @var{x}) -@end example - -@noindent -where @var{x} is a scalar value. If called as - -@example -randn ("seed") -@end example - -@noindent -@code{randn} returns the current value of the seed. -@end deftypefn +@DOCSTRING(randn) The @code{rand} and @code{randn} functions use separate generators. This ensures that @@ -544,23 +163,7 @@ Biomathematics at The University of Texas, M.D. Anderson Cancer Center, Houston, TX 77030. -@deftypefn {Built-in Function} {} diag (@var{v}, @var{k}) -Return a diagonal matrix with vector @var{v} on diagonal @var{k}. The -second argument is optional. If it is positive, the vector is placed on -the @var{k}-th super-diagonal. If it is negative, it is placed on the -@var{-k}-th sub-diagonal. The default value of @var{k} is 0, and the -vector is placed on the main diagonal. For example, - -@example -@group -diag ([1, 2, 3], 1) - @result{} 0 1 0 0 - 0 0 2 0 - 0 0 0 3 - 0 0 0 0 -@end group -@end example -@end deftypefn +@DOCSTRING(diag) @c XXX FIXME XXX -- is this really worth documenting? @c @@ -580,213 +183,25 @@ create vectors with evenly or logarithmically spaced elements. @xref{Ranges}. -@deftypefn {Function File} {} linspace (@var{base}, @var{limit}, @var{n}) -Return a row vector with @var{n} linearly spaced elements between -@var{base} and @var{limit}. The number of elements, @var{n}, must be -greater than 1. The @var{base} and @var{limit} are always included in -the range. If @var{base} is greater than @var{limit}, the elements are -stored in decreasing order. If the number of points is not specified, a -value of 100 is used. - -The @code{linspace} function always returns a row vector, regardless of -the value of @code{prefer_column_vectors}. -@end deftypefn - -@deftypefn {Function File} {} logspace (@var{base}, @var{limit}, @var{n}) -Similar to @code{linspace} except that the values are logarithmically -spaced from -@iftex -@tex -$10^{base}$ to $10^{limit}$. -@end tex -@end iftex -@ifinfo -10^base to 10^limit. -@end ifinfo +@DOCSTRING(linspace) -If @var{limit} is equal to -@iftex -@tex -$\pi$, -@end tex -@end iftex -@ifinfo -pi, -@end ifinfo -the points are between -@iftex -@tex -$10^{base}$ and $\pi$, -@end tex -@end iftex -@ifinfo -10^base and pi, -@end ifinfo -@emph{not} -@iftex -@tex -$10^{base}$ and $10^{\pi}$, -@end tex -@end iftex -@ifinfo -10^base and 10^pi, -@end ifinfo -in order to be compatible with the corresponding @sc{Matlab} function. -@end deftypefn +@DOCSTRING(logspace) -@defvr {Built-in Variable} treat_neg_dim_as_zero -If the value of @code{treat_neg_dim_as_zero} is nonzero, expressions -like - -@example -eye (-1) -@end example - -@noindent -produce an empty matrix (i.e., row and column dimensions are zero). -Otherwise, an error message is printed and control is returned to the -top level. The default value is 0. -@end defvr +@DOCSTRING(treat_neg_dim_as_zero) @node Famous Matrices, , Special Utility Matrices, Matrix Manipulation @section Famous Matrices The following functions return famous matrix forms. -@deftypefn {Function File} {} hankel (@var{c}, @var{r}) -Return the Hankel matrix constructed given the first column @var{c}, and -(optionally) the last row @var{r}. If the last element of @var{c} is -not the same as the first element of @var{r}, the last element of -@var{c} is used. If the second argument is omitted, the last row is -taken to be the same as the first column. - -A Hankel matrix formed from an m-vector @var{c}, and an n-vector -@var{r}, has the elements -@iftex -@tex -$$ -H (i, j) = \cases{c_{i+j-1},&$i+j-1\le m$;\cr r_{i+j-m},&otherwise.\cr} -$$ -@end tex -@end iftex -@ifinfo - -@example -@group -H (i, j) = c (i+j-1), i+j-1 <= m; -H (i, j) = r (i+j-m), otherwise -@end group -@end example -@end ifinfo -@end deftypefn +@DOCSTRING(hankel) -@deftypefn {Function File} {} hilb (@var{n}) -Return the Hilbert matrix of order @var{n}. The -@iftex -@tex -$i,\,j$ -@end tex -@end iftex -@ifinfo -i, j -@end ifinfo -element of a Hilbert matrix is defined as -@iftex -@tex -$$ -H (i, j) = {1 \over (i + j - 1)} -$$ -@end tex -@end iftex -@ifinfo +@DOCSTRING(hilb) -@example -H (i, j) = 1 / (i + j - 1) -@end example -@end ifinfo -@end deftypefn - -@deftypefn {Function File} {} invhilb (@var{n}) -Return the inverse of a Hilbert matrix of order @var{n}. This is exact. -Compare with the numerical calculation of @code{inverse (hilb (n))}, -which suffers from the ill-conditioning of the Hilbert matrix, and the -finite precision of your computer's floating point arithmetic. -@end deftypefn +@DOCSTRING(invhilb) -@deftypefn {Function File} {} sylvester_matrix (@var{k}) -Return the Sylvester matrix of order -@iftex -@tex -$n = 2^k$. -@end tex -@end iftex -@ifinfo -n = 2^k. -@end ifinfo -@end deftypefn - -@deftypefn {Function File} {} toeplitz (@var{c}, @var{r}) -Return the Toeplitz matrix constructed given the first column @var{c}, -and (optionally) the first row @var{r}. If the first element of @var{c} -is not the same as the first element of @var{r}, the first element of -@var{c} is used. If the second argument is omitted, the first row is -taken to be the same as the first column. - -A square Toeplitz matrix has the form -@iftex -@tex -$$ -\left[\matrix{c_0 & r_1 & r_2 & \ldots & r_n\cr - c_1 & c_0 & r_1 & & c_{n-1}\cr - c_2 & c_1 & c_0 & & c_{n-2}\cr - \vdots & & & & \vdots\cr - c_n & c_{n-1} & c_{n-2} & \ldots & c_0}\right]. -$$ -@end tex -@end iftex -@ifinfo +@DOCSTRING(sylvester_matrix) -@example -@group -c(0) r(1) r(2) ... r(n) -c(1) c(0) r(1) r(n-1) -c(2) c(1) c(0) r(n-2) - . . - . . - . . - -c(n) c(n-1) c(n-2) ... c(0) -@end group -@end example -@end ifinfo -@end deftypefn - -@deftypefn {Function File} {} vander (@var{c}) -Return the Vandermonde matrix whose next to last column is @var{c}. +@DOCSTRING(toeplitz) -A Vandermonde matrix has the form -@iftex -@tex -$$ -\left[\matrix{c_0^n & \ldots & c_0^2 & c_0 & 1\cr - c_1^n & \ldots & c_1^2 & c_1 & 1\cr - \vdots & & \vdots & \vdots & \vdots\cr - c_n^n & \ldots & c_n^2 & c_n & 1}\right]. -$$ -@end tex -@end iftex -@ifinfo - -@example -@group -c(0)^n ... c(0)^2 c(0) 1 -c(1)^n ... c(1)^2 c(1) 1 - . . . . - . . . . - . . . . - -c(n)^n ... c(n)^2 c(n) 1 -@end group -@end example -@end ifinfo -@end deftypefn +@DOCSTRING(vander)