Mercurial > hg > octave-lyh
changeset 9066:be150a172010
Cleanup documentation for diagperm.texi, sparse.texi
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| author | Rik <rdrider0-list@yahoo.com> |
|---|---|
| date | Sat, 28 Mar 2009 21:29:08 -0700 |
| parents | 8207b833557f |
| children | 8970b4b10e9f |
| files | doc/interpreter/diagperm.txi doc/interpreter/sparse.txi scripts/sparse/spaugment.m scripts/sparse/svds.m src/DLD-FUNCTIONS/amd.cc src/DLD-FUNCTIONS/symrcm.cc src/data.cc |
| diffstat | 7 files changed, 215 insertions(+), 167 deletions(-) [+] |
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--- a/doc/interpreter/diagperm.txi +++ b/doc/interpreter/diagperm.txi @@ -45,7 +45,7 @@ matrix. A permutation matrix is defined as a square matrix that has a single element equal to unity -in each row and each column; all other elements are zero. That is, there exists a +in each row and each column; all other elements are zero. That is, there exists a permutation (vector) @iftex @tex @@ -59,7 +59,7 @@ @end ifnottex Octave provides special treatment of real and complex rectangular diagonal matrices, -as well as permutation matrices. They are stored as special objects, using efficient +as well as permutation matrices. They are stored as special objects, using efficient storage and algorithms, facilitating writing both readable and efficient matrix algebra expressions in the Octave language. @@ -73,7 +73,7 @@ @subsection Creating Diagonal Matrices The most common and easiest way to create a diagonal matrix is using the built-in -function @dfn{diag}. The expression @code{diag (v)}, with @var{v} a vector, +function @dfn{diag}. The expression @code{diag (v)}, with @var{v} a vector, will create a square diagonal matrix with elements on the main diagonal given by the elements of @var{v}, and size equal to the length of @var{v}. @code{diag (v, m, n)} can be used to construct a rectangular diagonal matrix. @@ -81,11 +81,12 @@ than a general matrix object. Diagonal matrix with unit elements can be created using @dfn{eye}. -Some other built-in functions can also return diagonal matrices. Examples include +Some other built-in functions can also return diagonal matrices. Examples include @dfn{balance} or @dfn{inv}. Example: @example +@group diag (1:4) @result{} Diagonal Matrix @@ -105,6 +106,7 @@ 0 0 3 0 0 0 0 0 0 +@end group @end example @node Creating Permutation Matrices @@ -128,6 +130,7 @@ For example: @example +@group eye (4) ([1,3,2,4],:) @result{} Permutation Matrix @@ -145,16 +148,18 @@ 0 0 1 0 0 1 0 0 0 0 0 1 +@end group @end example Mathematically, an identity matrix is both diagonal and permutation matrix. In Octave, @code{eye (n)} returns a diagonal matrix, because a matrix -can only have one class. You can convert this diagonal matrix to a permutation +can only have one class. You can convert this diagonal matrix to a permutation matrix by indexing it by an identity permutation, as shown below. This is a special property of the identity matrix; indexing other diagonal matrices generally produces a full matrix. @example +@group eye (3) @result{} Diagonal Matrix @@ -170,21 +175,22 @@ 1 0 0 0 1 0 0 0 1 +@end group @end example -Some other built-in functions can also return permutation matrices. Examples include +Some other built-in functions can also return permutation matrices. Examples include @dfn{inv} or @dfn{lu}. @node Explicit and Implicit Conversions @subsection Explicit and Implicit Conversions -The diagonal and permutation matrices are special objects in their own right. A number +The diagonal and permutation matrices are special objects in their own right. A number of operations and built-in functions are defined for these matrices to use special, -more efficient code than would be used for a full matrix in the same place. Examples +more efficient code than would be used for a full matrix in the same place. Examples are given in further sections. To facilitate smooth mixing with full matrices, backward compatibility, and -compatibility with Matlab, the diagonal and permutation matrices should allow +compatibility with @sc{matlab}, the diagonal and permutation matrices should allow any operation that works on full matrices, and will either treat it specially, or implicitly convert themselves to full matrices. @@ -193,7 +199,7 @@ such as @dfn{exp}. An explicit conversion to a full matrix can be requested using the built-in -function @dfn{full}. It should also be noted that the diagonal and permutation +function @dfn{full}. It should also be noted that the diagonal and permutation matrix objects will cache the result of the conversion after it is first requested (explicitly or implicitly), so that subsequent conversions will be very cheap. @@ -203,7 +209,7 @@ As has been already said, diagonal and permutation matrices make it possible to use efficient algorithms while preserving natural linear -algebra syntax. This section describes in detail the operations that +algebra syntax. This section describes in detail the operations that are treated specially when performed on these special matrix objects. @menu @@ -214,8 +220,8 @@ @node Expressions Involving Diagonal Matrices @subsection Expressions Involving Diagonal Matrices -Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix, -then @code{D*M} will scale the rows of @var{M}. That means, +Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix, +then @code{D*M} will scale the rows of @var{M}. That means, if @code{S = D*M}, then for each pair of indices i,j it holds @iftex @@ -235,77 +241,79 @@ @example D(:,1:m) * M(1:m,:), @end example -i.e. trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n}, +i.e., trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n}, then @code{D*M} is equivalent to @example [D(1:n,n) * M; zeros(m-n, columns (M))], @end example -i.e. null rows are appended to the result. +i.e., null rows are appended to the result. The situation for right-multiplication @code{M*D} is analogous. The expressions @code{D \ M} and @code{M / D} perform inverse scaling. They are equivalent to solving a diagonal (or rectangular diagonal) -in a least-squares minimum-norm sense. In exact arithmetics, this is -equivalent to multiplying by a pseudoinverse. The pseudoinverse of +in a least-squares minimum-norm sense. In exact arithmetics, this is +equivalent to multiplying by a pseudoinverse. The pseudoinverse of a rectangular diagonal matrix is again a rectangular diagonal matrix with swapped dimensions, where each nonzero diagonal element is replaced by its reciprocal. The matrix division algorithms do, in fact, use division rather than multiplication by reciprocals for better numerical accuracy; otherwise, they honor the above definition. Note that a diagonal matrix is never truncated due -to ill-conditioning; otherwise, it would not be much useful for scaling. This -is typically consistent with linear algebra needs. A full matrix that only +to ill-conditioning; otherwise, it would not be much useful for scaling. This +is typically consistent with linear algebra needs. A full matrix that only happens to be diagonal (an is thus not a special object) is of course treated normally. Multiplication and division by diagonal matrices works efficiently also when -combined with sparse matrices, i.e. @code{D*S}, where @var{D} is a diagonal +combined with sparse matrices, i.e., @code{D*S}, where @var{D} is a diagonal matrix and @var{S} is a sparse matrix scales the rows of the sparse matrix and -returns a sparse matrix. The expressions @code{S*D}, @code{D\S}, @code{S/D} work +returns a sparse matrix. The expressions @code{S*D}, @code{D\S}, @code{S/D} work analogically. If @var{D1} and @var{D2} are both diagonal matrices, then the expressions @example +@group D1 + D2 D1 - D2 D1 * D2 D1 / D2 D1 \ D2 +@end group @end example again produce diagonal matrices, provided that normal -dimension matching rules are obeyed. The relations used are same as described above. +dimension matching rules are obeyed. The relations used are same as described above. Also, a diagonal matrix @var{D} can be multiplied or divided by a scalar, or raised to a scalar power if it is square, producing diagonal matrix result in all cases. A diagonal matrix can also be transposed or conjugate-transposed, giving the expected -result. Extracting a leading submatrix of a diagonal matrix, i.e. @code{D(1:m,1:n)}, +result. Extracting a leading submatrix of a diagonal matrix, i.e., @code{D(1:m,1:n)}, will produce a diagonal matrix, other indexing expressions will implicitly convert to full matrix. -Adding a diagonal matrix to a full matrix only operates on the diagonal elements. Thus, +Adding a diagonal matrix to a full matrix only operates on the diagonal elements. Thus, @example A = A + eps * eye (n) @end example -is an efficient method of augmenting the diagonal of a matrix. Subtraction works +is an efficient method of augmenting the diagonal of a matrix. Subtraction works analogically. When involved in expressions with other element-by-element operators, @code{.*}, @code{./}, @code{.\} or @code{.^}, an implicit conversion to full matrix will -take place. This is not always strictly necessary but chosen to facilitate -better consistency with Matlab. +take place. This is not always strictly necessary but chosen to facilitate +better consistency with @sc{matlab}. @node Expressions Involving Permutation Matrices @subsection Expressions Involving Permutation Matrices If @var{P} is a permutation matrix and @var{M} a matrix, the expression -@code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will +@code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will yield a column permutation. Matrix division @code{P\M} and @code{M/P} can be used to do inverse permutation. The previously described syntax for creating permutation matrices can actually help an user to understand the connection between a permutation matrix and -a permuting vector. Namely, the following holds, where @code{I = eye (n)} +a permuting vector. Namely, the following holds, where @code{I = eye (n)} is an identity matrix: @example I(p,:) * M = (I*M) (p,:) = M(p,:) @@ -320,20 +328,20 @@ A permutation matrix can be transposed (or conjugate-transposed, which is the same, because a permutation matrix is never complex), inverting the permutation, or equivalently, turning a row-permutation matrix into a -column-permutation one. For permutation matrices, transpose is equivalent to +column-permutation one. For permutation matrices, transpose is equivalent to inversion, thus @code{P\M} is equivalent to @code{P'*M}. Transpose of a permutation matrix (or inverse) is a constant-time operation, flipping only a flag internally, and thus the choice between the two above equivalent expressions for inverse permuting is completely up to the user's taste. Multiplication and division by permutation matrices works efficiently also when -combined with sparse matrices, i.e. @code{P*S}, where @var{P} is a permutation +combined with sparse matrices, i.e., @code{P*S}, where @var{P} is a permutation matrix and @var{S} is a sparse matrix permutes the rows of the sparse matrix and -returns a sparse matrix. The expressions @code{S*P}, @code{P\S}, @code{S/P} work +returns a sparse matrix. The expressions @code{S*P}, @code{P\S}, @code{S/P} work analogically. Two permutation matrices can be multiplied or divided (if their sizes match), performing -a composition of permutations. Also a permutation matrix can be indexed by a permutation +a composition of permutations. Also a permutation matrix can be indexed by a permutation vector (or two vectors), giving again a permutation matrix. Any other operations do not generally yield a permutation matrix and will thus trigger the implicit conversion. @@ -342,7 +350,7 @@ @section Functions That Are Aware of These Matrices This section lists the built-in functions that are aware of diagonal and -permutation matrices on input, or can return them as output. Passed to other +permutation matrices on input, or can return them as output. Passed to other functions, these matrices will in general trigger an implicit conversion. (Of course, user-defined dynamically linked functions may also work with diagonal or permutation matrices). @@ -356,7 +364,7 @@ @subsection Diagonal Matrix Functions @dfn{inv} and @dfn{pinv} can be applied to a diagonal matrix, yielding again -a diagonal matrix. @dfn{det} will use an efficient straightforward calculation +a diagonal matrix. @dfn{det} will use an efficient straightforward calculation when given a diagonal matrix, as well as @dfn{cond}. The following mapper functions can be applied to a diagonal matrix without converting it to a full one: @@ -370,7 +378,7 @@ @subsection Permutation Matrix Functions @dfn{inv} and @dfn{pinv} will invert a permutation matrix, preserving its -specialness. @dfn{det} can be applied to a permutation matrix, efficiently +specialness. @dfn{det} can be applied to a permutation matrix, efficiently calculating the sign of the permutation (which is equal to the determinant). A permutation matrix can also be returned from the built-in functions @@ -387,20 +395,24 @@ The following can be used to solve a linear system @code{A*x = b} using the pivoted LU factorization: @example +@group [L, U, P] = lu (A); ## now L*U = P*A x = U \ L \ P*b; +@end group @end example @noindent This is how you normalize columns of a matrix @var{X} to unit norm: @example +@group s = norm (X, "columns"); X = diag (s) \ X; +@end group @end example @noindent The following expression is a way to efficiently calculate the sign of a -permutation, given by a permutation vector @var{p}. It will also work +permutation, given by a permutation vector @var{p}. It will also work in earlier versions of Octave, but slowly. @example det (eye (length (p))(p, :)) @@ -410,10 +422,11 @@ Finally, here's how you solve a linear system @code{A*x = b} with Tikhonov regularization (ridge regression) using SVD (a skeleton only): @example +@group m = rows (A); n = columns (A); [U, S, V] = svd (A); ## determine the regularization factor alpha - ## alpha = ... + ## alpha = @dots{} ## transform to orthogonal basis b = U'*b; ## Use the standard formula, replacing A with S. @@ -421,6 +434,7 @@ x = (S'*S + alpha^2 * eye (n)) \ (S' * b); ## transform to solution basis x = V*x; +@end group @end example @@ -439,8 +453,8 @@ Numerical software dealing with structured and sparse matrices (including Octave) however, almost always makes a distinction between a "numerical zero" and an "assumed zero". -A "numerical zero" is a zero value occuring in a place where any floating-point -value could occur. It is normally stored somewhere in memory as an explicit +A "numerical zero" is a zero value occurring in a place where any floating-point +value could occur. It is normally stored somewhere in memory as an explicit value. An "assumed zero", on the contrary, is a zero matrix element implied by the matrix structure (diagonal, triangular) or a sparsity pattern; its value is @@ -453,7 +467,7 @@ or divided by @code{NaN}. The reason for this behavior is that the numerical multiplication is not actually performed anywhere by the underlying algorithm; the result is -just assumed to be zero. Equivalently, one can say that the part of the +just assumed to be zero. Equivalently, one can say that the part of the computation involving assumed zeros is performed symbolically, not numerically. This behavior not only facilitates the most straightforward and efficient @@ -469,11 +483,12 @@ Note that certain competing software does not strictly follow this principle and converts assumed zeros to numerical zeros in certain cases, while not doing -so in other cases. As of today, there are no intentions to mimick such behavior +so in other cases. As of today, there are no intentions to mimic such behavior in Octave. Examples of effects of assumed zeros vs. numerical zeros: @example +@group Inf * eye (3) @result{} Inf 0 0 @@ -494,9 +509,11 @@ NaN Inf NaN NaN NaN Inf +@end group @end example @example +@group diag(1:3) * [NaN; 1; 1] @result{} NaN @@ -513,5 +530,6 @@ NaN NaN NaN +@end group @end example
--- a/doc/interpreter/sparse.txi +++ b/doc/interpreter/sparse.txi @@ -37,21 +37,21 @@ @section The Creation and Manipulation of Sparse Matrices The size of mathematical problems that can be treated at any particular -time is generally limited by the available computing resources. Both, +time is generally limited by the available computing resources. Both, the speed of the computer and its available memory place limitation on the problem size. There are many classes of mathematical problems which give rise to -matrices, where a large number of the elements are zero. In this case +matrices, where a large number of the elements are zero. In this case it makes sense to have a special matrix type to handle this class of problems where only the non-zero elements of the matrix are -stored. Not only does this reduce the amount of memory to store the +stored. Not only does this reduce the amount of memory to store the matrix, but it also means that operations on this type of matrix can take advantage of the a-priori knowledge of the positions of the non-zero elements to accelerate their calculations. A matrix type that stores only the non-zero elements is generally called -sparse. It is the purpose of this document to discuss the basics of the +sparse. It is the purpose of this document to discuss the basics of the storage and creation of sparse matrices and the fundamental operations on them. @@ -66,15 +66,15 @@ @subsection Storage of Sparse Matrices It is not strictly speaking necessary for the user to understand how -sparse matrices are stored. However, such an understanding will help -to get an understanding of the size of sparse matrices. Understanding +sparse matrices are stored. However, such an understanding will help +to get an understanding of the size of sparse matrices. Understanding the storage technique is also necessary for those users wishing to create their own oct-files. -There are many different means of storing sparse matrix data. What all +There are many different means of storing sparse matrix data. What all of the methods have in common is that they attempt to reduce the complexity and storage given a-priori knowledge of the particular class of problems -that will be solved. A good summary of the available techniques for storing +that will be solved. A good summary of the available techniques for storing sparse matrix is given by Saad @footnote{Youcef Saad "SPARSKIT: A basic toolkit for sparse matrix computation", 1994, @url{http://www-users.cs.umn.edu/~saad/software/SPARSKIT/paper.ps}}. @@ -85,33 +85,35 @@ An obvious way to do this is by storing the elements of the matrix as triplets, with two elements being their position in the array -(rows and column) and the third being the data itself. This is conceptually +(rows and column) and the third being the data itself. This is conceptually easy to grasp, but requires more storage than is strictly needed. The storage technique used within Octave is the compressed column format. In this format the position of each element in a row and the -data are stored as previously. However, if we assume that all elements +data are stored as previously. However, if we assume that all elements in the same column are stored adjacent in the computers memory, then we only need to store information on the number of non-zero elements -in each column, rather than their positions. Thus assuming that the +in each column, rather than their positions. Thus assuming that the matrix has more non-zero elements than there are columns in the matrix, we win in terms of the amount of memory used. In fact, the column index contains one more element than the number of -columns, with the first element always being zero. The advantage of +columns, with the first element always being zero. The advantage of this is a simplification in the code, in that there is no special case -for the first or last columns. A short example, demonstrating this in +for the first or last columns. A short example, demonstrating this in C is. @example +@group for (j = 0; j < nc; j++) for (i = cidx (j); i < cidx(j+1); i++) printf ("non-zero element (%i,%i) is %d\n", ridx(i), j, data(i)); +@end group @end example A clear understanding might be had by considering an example of how the -above applies to an example matrix. Consider the matrix +above applies to an example matrix. Consider the matrix @example @group @@ -133,7 +135,7 @@ @end example This will be stored as three vectors @var{cidx}, @var{ridx} and @var{data}, -representing the column indexing, row indexing and data respectively. The +representing the column indexing, row indexing and data respectively. The contents of these three vectors for the above matrix will be @example @@ -146,22 +148,22 @@ Note that this is the representation of these elements with the first row and column assumed to start at zero, while in Octave itself the row and -column indexing starts at one. Thus the number of elements in the +column indexing starts at one. Thus the number of elements in the @var{i}-th column is given by @code{@var{cidx} (@var{i} + 1) - @var{cidx} (@var{i})}. Although Octave uses a compressed column format, it should be noted -that compressed row formats are equally possible. However, in the +that compressed row formats are equally possible. However, in the context of mixed operations between mixed sparse and dense matrices, it makes sense that the elements of the sparse matrices are in the -same order as the dense matrices. Octave stores dense matrices in +same order as the dense matrices. Octave stores dense matrices in column major ordering, and so sparse matrices are equally stored in this manner. A further constraint on the sparse matrix storage used by Octave is that all elements in the rows are stored in increasing order of their row -index, which makes certain operations faster. However, it imposes -the need to sort the elements on the creation of sparse matrices. Having +index, which makes certain operations faster. However, it imposes +the need to sort the elements on the creation of sparse matrices. Having disordered elements is potentially an advantage in that it makes operations such as concatenating two sparse matrices together easier and faster, however it adds complexity and speed problems elsewhere. @@ -173,11 +175,11 @@ @table @asis @item Returned from a function -There are many functions that directly return sparse matrices. These include +There are many functions that directly return sparse matrices. These include @dfn{speye}, @dfn{sprand}, @dfn{diag}, etc. @item Constructed from matrices or vectors The function @dfn{sparse} allows a sparse matrix to be constructed from -three vectors representing the row, column and data. Alternatively, the +three vectors representing the row, column and data. Alternatively, the function @dfn{spconvert} uses a three column matrix format to allow easy importation of data from elsewhere. @item Created and then filled @@ -188,15 +190,15 @@ @end table There are several basic functions to return specific sparse -matrices. For example the sparse identity matrix, is a matrix that is -often needed. It therefore has its own function to create it as +matrices. For example the sparse identity matrix, is a matrix that is +often needed. It therefore has its own function to create it as @code{speye (@var{n})} or @code{speye (@var{r}, @var{c})}, which creates an @var{n}-by-@var{n} or @var{r}-by-@var{c} sparse identity matrix. Another typical sparse matrix that is often needed is a random distribution -of random elements. The functions @dfn{sprand} and @dfn{sprandn} perform -this for uniform and normal random distributions of elements. They have exactly +of random elements. The functions @dfn{sprand} and @dfn{sprandn} perform +this for uniform and normal random distributions of elements. They have exactly the same calling convention, where @code{sprand (@var{r}, @var{c}, @var{d})}, creates an @var{r}-by-@var{c} sparse matrix with a density of filled elements of @var{d}. @@ -204,7 +206,7 @@ Other functions of interest that directly create sparse matrices, are @dfn{diag} or its generalization @dfn{spdiags}, that can take the definition of the diagonals of the matrix and create the sparse matrix -that corresponds to this. For example +that corresponds to this. For example @example s = diag (sparse(randn(1,n)), -1); @@ -234,9 +236,10 @@ The recommended way for the user to create a sparse matrix, is to create two vectors containing the row and column index of the data and a third -vector of the same size containing the data to be stored. For example +vector of the same size containing the data to be stored. For example @example +@group ri = ci = d = []; for j = 1:c ri = [ri; randperm(r)(1:n)']; @@ -244,32 +247,36 @@ d = [d; rand(n,1)]; endfor s = sparse (ri, ci, d, r, c); +@end group @end example creates an @var{r}-by-@var{c} sparse matrix with a random distribution -of @var{n} (<@var{r}) elements per column. The elements of the vectors +of @var{n} (<@var{r}) elements per column. The elements of the vectors do not need to be sorted in any particular order as Octave will sort -them prior to storing the data. However, pre-sorting the data will +them prior to storing the data. However, pre-sorting the data will make the creation of the sparse matrix faster. The function @dfn{spconvert} takes a three or four column real matrix. The first two columns represent the row and column index respectively and the third and four columns, the real and imaginary parts of the sparse -matrix. The matrix can contain zero elements and the elements can be -sorted in any order. Adding zero elements is a convenient way to define -the size of the sparse matrix. For example +matrix. The matrix can contain zero elements and the elements can be +sorted in any order. Adding zero elements is a convenient way to define +the size of the sparse matrix. For example @example +@group s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]') @result{} Compressed Column Sparse (rows=4, cols=4, nnz=3) (1 , 1) -> 1 (2 , 3) -> 2 (3 , 4) -> 3 +@end group @end example An example of creating and filling a matrix might be @example +@group k = 5; nz = r * k; s = spalloc (r, c, nz) @@ -278,13 +285,14 @@ s (:, j) = [zeros(r - k, 1); ... rand(k, 1)] (idx); endfor +@end group @end example It should be noted, that due to the way that the Octave assignment functions are written that the assignment will reallocate the memory used by the sparse matrix at each iteration of the above loop. Therefore the @dfn{spalloc} function ignores the @var{nz} argument and -does not preassign the memory for the matrix. Therefore, it is vitally +does not preassign the memory for the matrix. Therefore, it is vitally important that code using to above structure should be vectorized as much as possible to minimize the number of assignments and reduce the number of memory allocations. @@ -298,7 +306,7 @@ @DOCSTRING(spconvert) The above problem of memory reallocation can be avoided in -oct-files. However, the construction of a sparse matrix from an oct-file +oct-files. However, the construction of a sparse matrix from an oct-file is more complex than can be discussed here, and you are referred to chapter @ref{Dynamically Linked Functions}, to have a full description of the techniques involved. @@ -307,18 +315,18 @@ @subsection Finding out Information about Sparse Matrices There are a number of functions that allow information concerning -sparse matrices to be obtained. The most basic of these is +sparse matrices to be obtained. The most basic of these is @dfn{issparse} that identifies whether a particular Octave object is in fact a sparse matrix. Another very basic function is @dfn{nnz} that returns the number of non-zero entries there are in a sparse matrix, while the function @dfn{nzmax} returns the amount of storage allocated to the sparse -matrix. Note that Octave tends to crop unused memory at the first -opportunity for sparse objects. There are some cases of user created -sparse objects where the value returned by @dfn{nzmaz} will not be +matrix. Note that Octave tends to crop unused memory at the first +opportunity for sparse objects. There are some cases of user created +sparse objects where the value returned by @dfn{nzmax} will not be the same as @dfn{nnz}, but in general they will give the same -result. The function @dfn{spstats} returns some basic statistics on +result. The function @dfn{spstats} returns some basic statistics on the columns of a sparse matrix including the number of elements, the mean and the variance of each column. @@ -334,38 +342,42 @@ When solving linear equations involving sparse matrices Octave determines the means to solve the equation based on the type of the -matrix as discussed in @ref{Sparse Linear Algebra}. Octave probes the +matrix as discussed in @ref{Sparse Linear Algebra}. Octave probes the matrix type when the div (/) or ldiv (\) operator is first used with -the matrix and then caches the type. However the @dfn{matrix_type} +the matrix and then caches the type. However the @dfn{matrix_type} function can be used to determine the type of the sparse matrix prior -to use of the div or ldiv operators. For example +to use of the div or ldiv operators. For example @example +@group a = tril (sprandn(1024, 1024, 0.02), -1) ... + speye(1024); matrix_type (a); ans = Lower +@end group @end example show that Octave correctly determines the matrix type for lower -triangular matrices. @dfn{matrix_type} can also be used to force -the type of a matrix to be a particular type. For example +triangular matrices. @dfn{matrix_type} can also be used to force +the type of a matrix to be a particular type. For example @example +@group a = matrix_type (tril (sprandn (1024, ... 1024, 0.02), -1) + speye(1024), 'Lower'); +@end group @end example This allows the cost of determining the matrix type to be -avoided. However, incorrectly defining the matrix type will result in +avoided. However, incorrectly defining the matrix type will result in incorrect results from solutions of linear equations, and so it is entirely the responsibility of the user to correctly identify the matrix type There are several graphical means of finding out information about -sparse matrices. The first is the @dfn{spy} command, which displays +sparse matrices. The first is the @dfn{spy} command, which displays the structure of the non-zero elements of the -matrix. @xref{fig:spmatrix}, for an example of the use of +matrix. @xref{fig:spmatrix}, for an example of the use of @dfn{spy}. More advanced graphical information can be obtained with the @dfn{treeplot}, @dfn{etreeplot} and @dfn{gplot} commands. @@ -376,24 +388,26 @@ One use of sparse matrices is in graph theory, where the interconnections between nodes are represented as an adjacency -matrix. That is, if the i-th node in a graph is connected to the j-th -node. Then the ij-th node (and in the case of undirected graphs the -ji-th node) of the sparse adjacency matrix is non-zero. If each node -is then associated with a set of co-ordinates, then the @dfn{gplot} +matrix. That is, if the i-th node in a graph is connected to the j-th +node. Then the ij-th node (and in the case of undirected graphs the +ji-th node) of the sparse adjacency matrix is non-zero. If each node +is then associated with a set of coordinates, then the @dfn{gplot} command can be used to graphically display the interconnections between nodes. As a trivial example of the use of @dfn{gplot}, consider the example @example +@group A = sparse([2,6,1,3,2,4,3,5,4,6,1,5], [1,1,2,2,3,3,4,4,5,5,6,6],1,6,6); xy = [0,4,8,6,4,2;5,0,5,7,5,7]'; gplot(A,xy) +@end group @end example which creates an adjacency matrix @code{A} where node 1 is connected -to nodes 2 and 6, node 2 with nodes 1 and 3, etc. The co-ordinates of +to nodes 2 and 6, node 2 with nodes 1 and 3, etc. The coordinates of the nodes are given in the n-by-2 matrix @code{xy}. @ifset htmltex @xref{fig:gplot}. @@ -406,7 +420,7 @@ The dependencies between the nodes of a Cholesky factorization can be calculated in linear time without explicitly needing to calculate the -Cholesky factorization by the @code{etree} command. This command +Cholesky factorization by the @code{etree} command. This command returns the elimination tree of the matrix and can be displayed graphically by the command @code{treeplot(etree(A))} if @code{A} is symmetric or @code{treeplot(etree(A+A'))} otherwise. @@ -437,17 +451,17 @@ An important consideration in the use of the sparse functions of Octave is that many of the internal functions of Octave, such as -@dfn{diag}, cannot accept sparse matrices as an input. The sparse +@dfn{diag}, cannot accept sparse matrices as an input. The sparse implementation in Octave therefore uses the @dfn{dispatch} function to overload the normal Octave functions with equivalent -functions that work with sparse matrices. However, at any time the +functions that work with sparse matrices. However, at any time the sparse matrix specific version of the function can be used by explicitly calling its function name. The table below lists all of the sparse functions of Octave. Note that the names of the specific sparse forms of the functions are typically the same as -the general versions with a @dfn{sp} prefix. In the table below, and the +the general versions with a @dfn{sp} prefix. In the table below, and the rest of this article the specific sparse versions of the functions are used. @@ -487,9 +501,9 @@ @dfn{spparms}, @dfn{symbfact}, @dfn{spstats} @end table -In addition all of the standard Octave mapper functions (ie. basic +In addition all of the standard Octave mapper functions (i.e., basic math functions that take a single argument) such as @dfn{abs}, etc -can accept sparse matrices. The reader is referred to the documentation +can accept sparse matrices. The reader is referred to the documentation supplied with these functions within Octave itself for further details. @@ -497,22 +511,24 @@ @subsubsection The Return Types of Operators and Functions The two basic reasons to use sparse matrices are to reduce the memory -usage and to not have to do calculations on zero elements. The two are +usage and to not have to do calculations on zero elements. The two are closely related in that the computation time on a sparse matrix operator or function is roughly linear with the number of non-zero elements. Therefore, there is a certain density of non-zero elements of a matrix where it no longer makes sense to store it as a sparse matrix, but rather -as a full matrix. For this reason operators and functions that have a -high probability of returning a full matrix will always return one. For +as a full matrix. For this reason operators and functions that have a +high probability of returning a full matrix will always return one. For example adding a scalar constant to a sparse matrix will almost always make it a full matrix, and so the example @example +@group speye(3) + 0 @result{} 1 0 0 0 1 0 0 0 1 +@end group @end example returns a full matrix as can be seen. @@ -521,28 +537,28 @@ Additionally, if @code{sparse_auto_mutate} is true, all sparse functions test the amount of memory occupied by the sparse matrix to see if the amount of storage used is larger than the amount used by the full -equivalent. Therefore @code{speye (2) * 1} will return a full matrix as +equivalent. Therefore @code{speye (2) * 1} will return a full matrix as the memory used is smaller for the full version than the sparse version. As all of the mixed operators and functions between full and sparse -matrices exist, in general this does not cause any problems. However, +matrices exist, in general this does not cause any problems. However, one area where it does cause a problem is where a sparse matrix is promoted to a full matrix, where subsequent operations would resparsify -the matrix. Such cases are rare, but can be artificially created, for +the matrix. Such cases are rare, but can be artificially created, for example @code{(fliplr(speye(3)) + speye(3)) - speye(3)} gives a full -matrix when it should give a sparse one. In general, where such cases +matrix when it should give a sparse one. In general, where such cases occur, they impose only a small memory penalty. There is however one known case where this behavior of Octave's -sparse matrices will cause a problem. That is in the handling of the -@dfn{diag} function. Whether @dfn{diag} returns a sparse or full matrix -depending on the type of its input arguments. So +sparse matrices will cause a problem. That is in the handling of the +@dfn{diag} function. Whether @dfn{diag} returns a sparse or full matrix +depending on the type of its input arguments. So @example a = diag (sparse([1,2,3]), -1); @end example -should return a sparse matrix. To ensure this actually happens, the +should return a sparse matrix. To ensure this actually happens, the @dfn{sparse} function, and other functions based on it like @dfn{speye}, always returns a sparse matrix, even if the memory used will be larger than its full representation. @@ -550,19 +566,20 @@ @DOCSTRING(sparse_auto_mutate) Note that the @code{sparse_auto_mutate} option is incompatible with -@sc{Matlab}, and so it is off by default. +@sc{matlab}, and so it is off by default. @node Mathematical Considerations @subsubsection Mathematical Considerations The attempt has been made to make sparse matrices behave in exactly the -same manner as there full counterparts. However, there are certain differences +same manner as there full counterparts. However, there are certain differences and especially differences with other products sparse implementations. -Firstly, the "./" and ".^" operators must be used with care. Consider what +Firstly, the "./" and ".^" operators must be used with care. Consider what the examples @example +@group s = speye (4); a1 = s .^ 2; a2 = s .^ s; @@ -570,10 +587,11 @@ a4 = s ./ 2; a5 = 2 ./ s; a6 = s ./ s; +@end group @end example -will give. The first example of @var{s} raised to the power of 2 causes -no problems. However @var{s} raised element-wise to itself involves a +will give. The first example of @var{s} raised to the power of 2 causes +no problems. However @var{s} raised element-wise to itself involves a large number of terms @code{0 .^ 0} which is 1. There @code{@var{s} .^ @var{s}} is a full matrix. @@ -582,7 +600,7 @@ For the "./" operator @code{@var{s} ./ 2} has no problems, but @code{2 ./ @var{s}} involves a large number of infinity terms as well -and is equally a full matrix. The case of @code{@var{s} ./ @var{s}} +and is equally a full matrix. The case of @code{@var{s} ./ @var{s}} involves terms like @code{0 ./ 0} which is a @code{NaN} and so this is equally a full matrix with the zero elements of @var{s} filled with @code{NaN} values. @@ -592,9 +610,10 @@ A particular problem of sparse matrices comes about due to the fact that as the zeros are not stored, the sign-bit of these zeros is equally not -stored. In certain cases the sign-bit of zero is important. For example +stored. In certain cases the sign-bit of zero is important. For example @example +@group a = 0 ./ [-1, 1; 1, -1]; b = 1 ./ a @result{} -Inf Inf @@ -602,27 +621,28 @@ c = 1 ./ sparse (a) @result{} Inf Inf Inf Inf +@end group @end example To correct this behavior would mean that zero elements with a negative sign-bit would need to be stored in the matrix to ensure that their -sign-bit was respected. This is not done at this time, for reasons of +sign-bit was respected. This is not done at this time, for reasons of efficiency, and so the user is warned that calculations where the sign-bit of zero is important must not be done using sparse matrices. In general any function or operator used on a sparse matrix will result in a sparse matrix with the same or a larger number of non-zero -elements than the original matrix. This is particularly true for the -important case of sparse matrix factorizations. The usual way to +elements than the original matrix. This is particularly true for the +important case of sparse matrix factorizations. The usual way to address this is to reorder the matrix, such that its factorization is -sparser than the factorization of the original matrix. That is the +sparser than the factorization of the original matrix. That is the factorization of @code{L * U = P * S * Q} has sparser terms @code{L} and @code{U} than the equivalent factorization @code{L * U = S}. Several functions are available to reorder depending on the type of the -matrix to be factorized. If the matrix is symmetric positive-definite, -then @dfn{symamd} or @dfn{csymamd} should be used. Otherwise -@dfn{amd}, @dfn{colamd} or @dfn{ccolamd} should be used. For completeness +matrix to be factorized. If the matrix is symmetric positive-definite, +then @dfn{symamd} or @dfn{csymamd} should be used. Otherwise +@dfn{amd}, @dfn{colamd} or @dfn{ccolamd} should be used. For completeness the reordering functions @dfn{colperm} and @dfn{randperm} are also available. @@ -636,7 +656,7 @@ The standard Cholesky factorization of this matrix can be obtained by the same command that would be used for a full -matrix. This can be visualized with the command +matrix. This can be visualized with the command @code{r = chol(A); spy(r);}. @ifset HAVE_CHOLMOD @ifset HAVE_COLAMD @@ -661,7 +681,7 @@ @ifset htmltex 10200, @end ifset -with only half of the symmetric matrix being stored. This +with only half of the symmetric matrix being stored. This is a significant level of fill in, and although not an issue for such a small test case, can represents a large overhead in working with other sparse matrices. @@ -669,7 +689,7 @@ The appropriate sparsity preserving permutation of the original matrix is given by @dfn{symamd} and the factorization using this reordering can be visualized using the command @code{q = symamd(A); -r = chol(A(q,q)); spy(r)}. This gives +r = chol(A(q,q)); spy(r)}. This gives @ifinfo @ifnothtml 29 @@ -729,7 +749,7 @@ Octave includes a polymorphic solver for sparse matrices, where the exact solver used to factorize the matrix, depends on the properties -of the sparse matrix itself. Generally, the cost of determining the matrix type +of the sparse matrix itself. Generally, the cost of determining the matrix type is small relative to the cost of factorizing the matrix itself, but in any case the matrix type is cached once it is calculated, so that it is not re-determined each time it is used in a linear equation. @@ -740,7 +760,7 @@ @item If the matrix is diagonal, solve directly and goto 8 @item If the matrix is a permuted diagonal, solve directly taking into -account the permutations. Goto 8 +account the permutations. Goto 8 @item If the matrix is square, banded and if the band density is less than that given by @code{spparms ("bandden")} continue, else goto 4. @@ -788,30 +808,30 @@ @end enumerate The band density is defined as the number of non-zero values in the matrix -divided by the number of non-zero values in the matrix. The banded matrix +divided by the number of non-zero values in the matrix. The banded matrix solvers can be entirely disabled by using @dfn{spparms} to set @code{bandden} -to 1 (i.e. @code{spparms ("bandden", 1)}). +to 1 (i.e., @code{spparms ("bandden", 1)}). -The QR solver factorizes the problem with a Dulmage-Mendhelsohn, to +The QR solver factorizes the problem with a Dulmage-Mendelsohn, to separate the problem into blocks that can be treated as over-determined, -multiple well determined blocks, and a final over-determined block. For +multiple well determined blocks, and a final over-determined block. For matrices with blocks of strongly connected nodes this is a big win as -LU decomposition can be used for many blocks. It also significantly +LU decomposition can be used for many blocks. It also significantly improves the chance of finding a solution to over-determined problems rather than just returning a vector of @dfn{NaN}'s. All of the solvers above, can calculate an estimate of the condition -number. This can be used to detect numerical stability problems in the -solution and force a minimum norm solution to be used. However, for +number. This can be used to detect numerical stability problems in the +solution and force a minimum norm solution to be used. However, for narrow banded, triangular or diagonal matrices, the cost of calculating the condition number is significant, and can in fact -exceed the cost of factoring the matrix. Therefore the condition +exceed the cost of factoring the matrix. Therefore the condition number is not calculated in these cases, and Octave relies on simpler techniques to detect singular matrices or the underlying LAPACK code in the case of banded matrices. The user can force the type of the matrix with the @code{matrix_type} -function. This overcomes the cost of discovering the type of the matrix. +function. This overcomes the cost of discovering the type of the matrix. However, it should be noted that identifying the type of the matrix incorrectly will lead to unpredictable results, and so @code{matrix_type} should be used with care. @@ -848,7 +868,7 @@ The left division @code{\} and right division @code{/} operators, discussed in the previous section, use direct solvers to resolve a linear equation of the form @code{@var{x} = @var{A} \ @var{b}} or -@code{@var{x} = @var{b} / @var{A}}. Octave equally includes a number of +@code{@var{x} = @var{b} / @var{A}}. Octave equally includes a number of functions to solve sparse linear equations using iterative techniques. @DOCSTRING(pcg) @@ -856,9 +876,9 @@ @DOCSTRING(pcr) The speed with which an iterative solver converges to a solution can be -accelerated with the use of a pre-conditioning matrix @var{M}. In this +accelerated with the use of a pre-conditioning matrix @var{M}. In this case the linear equation @code{@var{M}^-1 * @var{x} = @var{M}^-1 * -@var{A} \ @var{b}} is solved instead. Typical pre-conditioning matrices +@var{A} \ @var{b}} is solved instead. Typical pre-conditioning matrices are partial factorizations of the original matrix. @DOCSTRING(luinc) @@ -867,13 +887,13 @@ @section Real Life Example of the use of Sparse Matrices A common application for sparse matrices is in the solution of Finite -Element Models. Finite element models allow numerical solution of +Element Models. Finite element models allow numerical solution of partial differential equations that do not have closed form solutions, typically because of the complex shape of the domain. In order to motivate this application, we consider the boundary value -Laplace equation. This system can model scalar potential fields, such -as heat or electrical potential. Given a medium +Laplace equation. This system can model scalar potential fields, such +as heat or electrical potential. Given a medium @iftex @tex $\Omega$ @@ -925,7 +945,7 @@ @end ifinfo and know the boundary temperature (Dirichlet condition) or heat flux (from which we can calculate the Neumann condition -by dividing by the thermal conductivity at the boundary). Similarly, +by dividing by the thermal conductivity at the boundary). Similarly, in an electrical model, we want to calculate the voltage in @iftex @tex @@ -955,19 +975,20 @@ We take as an 3D example a cylindrical liquid filled tank with a small non-conductive ball from the EIDORS project@footnote{EIDORS - Electrical Impedance Tomography and Diffuse optical Tomography Reconstruction Software -@url{http://eidors3d.sourceforge.net}}. This is model is designed to reflect -an application of electrical impedance tomography, where current patterns -are applied to such a tank in order to image the internal conductivity -distribution. In order to describe the FEM geometry, we have a matrix of +@url{http://eidors3d.sourceforge.net}}. This is model is designed to reflect +an application of electrical impedance tomography, where current patterns +are applied to such a tank in order to image the internal conductivity +distribution. In order to describe the FEM geometry, we have a matrix of vertices @code{nodes} and simplices @code{elems}. @end ifset The following example creates a simple rectangular 2D electrically conductive medium with 10 V and 20 V imposed on opposite sides -(Dirichlet boundary conditions). All other edges are electrically +(Dirichlet boundary conditions). All other edges are electrically isolated. @example +@group node_y= [1;1.2;1.5;1.8;2]*ones(1,11); node_x= ones(5,1)*[1,1.05,1.1,1.2, ... 1.3,1.5,1.7,1.8,1.9,1.95,2]; @@ -985,20 +1006,23 @@ E= size(elems,1); # No. of simplices N= size(nodes,1); # No. of vertices D= size(elems,2); # dimensions+1 +@end group @end example This creates a N-by-2 matrix @code{nodes} and a E-by-3 matrix @code{elems} with values, which define finite element triangles: @example +@group nodes(1:7,:)' - 1.00 1.00 1.00 1.00 1.00 1.05 1.05 ... - 1.00 1.20 1.50 1.80 2.00 1.00 1.20 ... + 1.00 1.00 1.00 1.00 1.00 1.05 1.05 @dots{} + 1.00 1.20 1.50 1.80 2.00 1.00 1.20 @dots{} elems(1:7,:)' - 1 2 3 4 2 3 4 ... - 2 3 4 5 7 8 9 ... - 6 7 8 9 6 7 8 ... + 1 2 3 4 2 3 4 @dots{} + 2 3 4 5 7 8 9 @dots{} + 6 7 8 9 6 7 8 @dots{} +@end group @end example Using a first order FEM, we approximate the electrical conductivity @@ -1014,12 +1038,13 @@ as constant on each simplex (represented by the vector @code{conductivity}). Based on the finite element geometry, we first calculate a system (or stiffness) matrix for each simplex (represented as 3-by-3 elements on the -diagonal of the element-wise system matrix @code{SE}. Based on @code{SE} +diagonal of the element-wise system matrix @code{SE}. Based on @code{SE} and a N-by-DE connectivity matrix @code{C}, representing the connections between simplices and vertices, the global connectivity matrix @code{S} is calculated. @example +@group # Element conductivity conductivity= [1*ones(1,16), ... 2*ones(1,48), 1*ones(1,16)]; @@ -1046,6 +1071,7 @@ SE= sparse(Siidx,Sjidx,Sdata); # Global system matrix S= C'* SE *C; +@end group @end example The system matrix acts like the conductivity @@ -1070,6 +1096,7 @@ solve for the voltages at each vertex @code{V}. @example +@group # Dirichlet boundary conditions D_nodes=[1:5, 51:55]; D_value=[10*ones(1,5), 20*ones(1,5)]; @@ -1080,7 +1107,7 @@ # boundary condns idx(D_nodes) = []; - # Neumann boundary conditions. Note that + # Neumann boundary conditions. Note that # N_value must be normalized by the # boundary length and element conductivity N_nodes=[]; @@ -1091,6 +1118,7 @@ V(idx) = S(idx,idx) \ ( Q(idx) - ... S(idx,D_nodes) * V(D_nodes)); +@end group @end example Finally, in order to display the solution, we show each solved voltage @@ -1106,12 +1134,14 @@ @end ifset @example +@group elemx = elems(:,[1,2,3,1])'; xelems = reshape (nodes(elemx, 1), 4, E); yelems = reshape (nodes(elemx, 2), 4, E); velems = reshape (V(elemx), 4, E); plot3 (xelems,yelems,velems,'k'); print ('grid.eps'); +@end group @end example
--- a/scripts/sparse/spaugment.m +++ b/scripts/sparse/spaugment.m @@ -28,7 +28,7 @@ ## @end example ## ## @noindent -## This is related to the leasted squared solution of +## This is related to the least squares solution of ## @code{@var{a} \\ @var{b}}, by ## ## @example
--- a/scripts/sparse/svds.m +++ b/scripts/sparse/svds.m @@ -55,7 +55,7 @@ ## 1e-10. ## ## @item maxit -## The maximum number of iterations. The defaut is 300. +## The maximum number of iterations. The default is 300. ## ## @item disp ## The level of diagnostic printout. If @code{disp} is 0 then there is no
--- a/src/DLD-FUNCTIONS/amd.cc +++ b/src/DLD-FUNCTIONS/amd.cc @@ -67,14 +67,14 @@ @table @asis\n\ @item opts.dense\n\ Determines what @code{amd} considers to be a dense row or column of the\n\ -input matrix. Rows or columns with more that @code{max(16, (dense *\n\ +input matrix. Rows or columns with more than @code{max(16, (dense *\n\ sqrt (@var{n})} entries, where @var{n} is the order of the matrix @var{s},\n\ -are igorned by @code{amd} during the calculation of the permutation\n\ +are ignored by @code{amd} during the calculation of the permutation\n\ The value of dense must be a positive scalar and its default value is 10.0\n\ \n\ @item opts.aggressive\n\ -If this value is a non zero scalar, then @code{amd} performs agressive\n\ -absorption. The default is not to perform agressive absorption.\n\ +If this value is a non zero scalar, then @code{amd} performs aggressive\n\ +absorption. The default is not to perform aggressive absorption.\n\ @end table\n\ \n\ The author of the code itself is Timothy A. Davis (davis@@cise.ufl.edu),\n\
--- a/src/DLD-FUNCTIONS/symrcm.cc +++ b/src/DLD-FUNCTIONS/symrcm.cc @@ -430,7 +430,7 @@ descriptions found in\n\ \n\ E. Cuthill, J. McKee: Reducing the Bandwidth of Sparse Symmetric\n\ -Matrices. Proceedings of the 24th ACM National Conference, 157-172\n\ +Matrices. Proceedings of the 24th ACM National Conference, 157--172\n\ 1969, Brandon Press, New Jersey.\n\ \n\ Alan George, Joseph W. H. Liu: Computer Solution of Large Sparse\n\
--- a/src/data.cc +++ b/src/data.cc @@ -2493,7 +2493,7 @@ Return the amount of storage allocated to the sparse matrix @var{SM}.\n\ Note that Octave tends to crop unused memory at the first opportunity\n\ for sparse objects. There are some cases of user created sparse objects\n\ -where the value returned by @dfn{nzmaz} will not be the same as @dfn{nnz},\n\ +where the value returned by @dfn{nzmax} will not be the same as @dfn{nnz},\n\ but in general they will give the same result.\n\ @seealso{sparse, spalloc}\n\ @end deftypefn")