From: Denis Davydov Date: Sun, 25 Mar 2018 08:47:45 +0000 (+0200) Subject: doc: fix LaTeX in Lapack and ScaLapack X-Git-Tag: v9.0.0-rc1~257^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F6101%2Fhead;p=dealii.git doc: fix LaTeX in Lapack and ScaLapack --- diff --git a/include/deal.II/lac/lapack_full_matrix.h b/include/deal.II/lac/lapack_full_matrix.h index 26f57767f0..7d22371b0f 100644 --- a/include/deal.II/lac/lapack_full_matrix.h +++ b/include/deal.II/lac/lapack_full_matrix.h @@ -519,7 +519,7 @@ public: void compute_cholesky_factorization (); /** - * Estimate the reciprocal of the condition number $1/k(\mathbf A)$ in $L_1$ norm ($1/(||\mathbf A||_1 ||\mathbf A^{-1}||_1)$) + * Estimate the reciprocal of the condition number $1/k(\mathbf A)$ in $L_1$ norm ($1/(||\mathbf A||_1 \, ||\mathbf A^{-1}||_1)$) * of a symmetric positive definite matrix using Cholesky factorization. This function can only * be called if the matrix is already factorized. * @@ -650,16 +650,16 @@ public: /** * Compute eigenvalues and eigenvectors of a real symmetric matrix. Only - * eigenvalues in the interval $(lower_bound, upper_bound]$ are computed with - * the absolute tolerance abs_accuracy. An approximate eigenvalue is + * eigenvalues in the interval $(\rm{lower\_bound}, \rm{upper\_bound}]$ are computed with + * the absolute tolerance $\rm abs\_accuracy$. An approximate eigenvalue is * accepted as converged when it is determined to lie in an interval $[a,b]$ - * of width less than or equal to $abs_accuracy + eps * max(|a|,|b|)$, where - * $eps$ is the machine precision. If $abs_accuracy$ is less than or equal to - * zero, then $eps*|t|$ will be used in its place, where $|t|$ is the 1-norm of + * of width less than or equal to $\rm{abs\_accuracy} + eps * \rm{max}(|a|,|b|)$, where + * $eps$ is the machine precision. If $\rm{abs\_accuracy}$ is less than or equal to + * zero, then $eps\,|\mathbf{T}|_1$ will be used in its place, where $|\mathbf{T}|_1$ is the 1-norm of * the tridiagonal matrix obtained by reducing $\mathbf A$ to tridiagonal form. - * Eigenvalues will be computed most accurately when $abs_accuracy$ is set to + * Eigenvalues will be computed most accurately when $\rm{abs\_accuracy}$ is set to * twice the underflow threshold, not zero. After this routine has been - * called, all eigenvalues in $(lower_bound, upper_bound]$ will be stored in + * called, all eigenvalues in $(\rm{lower\_bound}, \rm{upper\_bound}]$ will be stored in * eigenvalues and the corresponding eigenvectors will be stored in the * columns of eigenvectors, whose dimension is set accordingly. * @@ -674,21 +674,22 @@ public: /** * Compute generalized eigenvalues and eigenvectors of a real generalized * symmetric eigenproblem of the form - * - itype = 1: $\mathbf A \cdot \mathbf x=\lambda \mathbf B \cdot x$ + * - itype = 1: $\mathbf A \cdot \mathbf x=\lambda \mathbf B \cdot \mathbf x$ * - itype = 2: $\mathbf A \cdot \mathbf B \cdot \mathbf x=\lambda \mathbf x$ * - itype = 3: $\mathbf B \cdot \mathbf A \cdot \mathbf x=\lambda \mathbf x$ + * * where $\mathbf A$ is this matrix. $\mathbf A$ * and $\mathbf B$ are assumed to be symmetric, and $\mathbf B$ has to be positive definite. - * Only eigenvalues in the interval $(lower_bound, upper_bound]$ are computed - * with the absolute tolerance $abs_accuracy$. An approximate eigenvalue is + * Only eigenvalues in the interval $(\rm{lower\_bound}, \rm{upper\_bound}]$ are computed + * with the absolute tolerance $\rm{abs\_accuracy}$. An approximate eigenvalue is * accepted as converged when it is determined to lie in an interval $[a,b]$ - * of width less than or equal to $abs_accuracy + eps * max( |a|,|b| )$, where - * $eps$ is the machine precision. If $abs_accuracy$ is less than or equal to - * zero, then $eps*|t|$ will be used in its place, where $|t|$ is the 1-norm of + * of width less than or equal to $\rm{abs\_accuracy} + eps * \rm{max}( |a|,|b| )$, where + * $eps$ is the machine precision. If $\rm{abs\_accuracy}$ is less than or equal to + * zero, then $eps \, |\mathbf{T}|_1$ will be used in its place, where $|\mathbf{T}|_1$ is the 1-norm of * the tridiagonal matrix obtained by reducing $\mathbf A$ to tridiagonal form. - * Eigenvalues will be computed most accurately when $abs_accuracy$ is set to + * Eigenvalues will be computed most accurately when $\rm{abs\_accuracy}$ is set to * twice the underflow threshold, not zero. After this routine has been - * called, all eigenvalues in $(lower_bound, upper_bound]$ will be stored in + * called, all eigenvalues in $(\rm{lower\_bound}, \rm{upper\_bound}]$ will be stored in * eigenvalues and the corresponding eigenvectors will be stored in * eigenvectors, whose dimension is set accordingly. * @@ -747,7 +748,7 @@ public: * The parameter @p threshold determines, when a singular value should * be considered zero. It is the ratio of the smallest to the largest * nonzero singular value $s_{max}$. Thus, the inverses of all - * singular values less than $s_{max}/threshold$ will + * singular values less than $s_{max}/\rm{threshold}$ will * be set to zero. */ void compute_inverse_svd (const double threshold = 0.); diff --git a/include/deal.II/lac/scalapack.h b/include/deal.II/lac/scalapack.h index 19dca92b6e..9a5d1e6b31 100644 --- a/include/deal.II/lac/scalapack.h +++ b/include/deal.II/lac/scalapack.h @@ -189,7 +189,8 @@ public: * * * If it is necessary to copy complete matrices with an identical block-cyclic distribution, - * use copy_to(ScaLAPACKMatrix &dest) with only one argument to avoid communication. + * use ScaLAPACKMatrix::copy_to(ScaLAPACKMatrix &dest) + * with only one argument to avoid communication. * * The underlying process grids of the matrices @p A and @p B must have been built * with the same MPI communicator. @@ -225,7 +226,7 @@ public: /** * Matrix-addition: - * $\mathbf{A} = \mathbf{A} + b \mathbf{B}$ + * $\mathbf{A} = \mathbf{A} + b\, \mathbf{B}$ * * The matrices $\mathbf{A}$ and $\mathbf{B}$ must have the same process grid. * @@ -236,7 +237,7 @@ public: /** * Matrix-addition: - * $\mathbf{A} = \mathbf{A} + b \mathbf{B}^T$ + * $\mathbf{A} = \mathbf{A} + b\, \mathbf{B}^T$ * * The matrices $\mathbf{A}$ and $\mathbf{B}$ must have the same process grid. * @@ -395,7 +396,7 @@ public: /** * Computing selected eigenvalues and, optionally, the eigenvectors of the real symmetric - * matrix $A \in \mathbb{R}^{M \times M}$. + * matrix $\mathbf{A} \in \mathbb{R}^{M \times M}$. * * The eigenvalues/eigenvectors are selected by prescribing a range of indices @p index_limits. * @@ -423,22 +424,22 @@ public: /** * Computing the singular value decomposition (SVD) of a - * matrix $A \in \mathbb{R}^{M \times N}$, optionally computing the left and/or right - * singular vectors. The SVD is written as $A = U * \Sigma * V^T$ - * with $\Sigma \in \mathbb{R}^{M \times N}$ as a diagonal matrix, - * $U \in \mathbb{R}^{M \times M}$ and $U \in \mathbb{R}^{M \times M}$ - * as orthogonal matrices. The diagonal elements of $\Sigma$ - * are the singular values of $A$ and the columns of $U$ and $V$ are the + * matrix $\mathbf{A} \in \mathbb{R}^{M \times N}$, optionally computing the left and/or right + * singular vectors. The SVD is written as $\mathbf{A} = \mathbf{U} \cdot \mathbf{\Sigma} \cdot \mathbf{V}^T$ + * with $\mathbf{\Sigma} \in \mathbb{R}^{M \times N}$ as a diagonal matrix, + * $\mathbf{U} \in \mathbb{R}^{M \times M}$ and $\mathbf{V} \in \mathbb{R}^{M \times M}$ + * as orthogonal matrices. The diagonal elements of $\mathbf{\Sigma}$ + * are the singular values of $A$ and the columns of $\mathbf{U}$ and $\mathbf{V}$ are the * corresponding left and right singular vectors, respectively. The * singular values are returned in decreasing order and only the first $\min(M,N)$ - * columns of $U$ and rows of VT = $V^T$ are computed. + * columns of $\mathbf{U}$ and rows of $\mathbf{V}^T$ are computed. * * Upon return the content of the matrix is unusable. - * The matrix A must have identical block cyclic distribution for the rows and column. + * The matrix $\mathbf{A}$ must have identical block cyclic distribution for the rows and column. * - * If left singular vectors are required matrices $A$ and $U$ + * If left singular vectors are required matrices $\mathbf{A}$ and $\mathbf{U}$ * have to be constructed with the same process grid and block cyclic distribution. - * If right singular vectors are required matrices $A$ and $V^T$ + * If right singular vectors are required matrices $\mathbf{A}$ and $\mathbf{V}^T$ * have to be constructed with the same process grid and block cyclic distribution. * * To avoid computing the left and/or right singular vectors the function accepts nullptr @@ -449,33 +450,33 @@ public: /** * Solving overdetermined or underdetermined real linear - * systems involving matrix $A \in \mathbb{R}^{M \times N}$, or its transpose $A^T$, - * using a QR or LQ factorization of $A$ for $N_{\rm RHS}$ RHS vectors in the columns of matrix $B$ + * systems involving matrix $\mathbf{A} \in \mathbb{R}^{M \times N}$, or its transpose $\mathbf{A}^T$, + * using a QR or LQ factorization of $\mathbf{A}$ for $N_{\rm RHS}$ RHS vectors in the columns of matrix $\mathbf{B}$ * - * It is assumed that $A$ has full rank: $rank(A) = \min(M,N)$. + * It is assumed that $\mathbf{A}$ has full rank: $\rm{rank}(\mathbf{A}) = \min(M,N)$. * * The following options are supported: * -# If(!transpose) and $M \geq N$: least squares solution of overdetermined system - * $\min \Vert B - A*X\Vert$.\n - * Upon exit the rows $0$ to $N-1$ of $B$ contain the least square solution vectors. The residual sum of squares + * $\min \Vert \mathbf{B} - \mathbf{A}\cdot \mathbf{X}\Vert$.\n + * Upon exit the rows $0$ to $N-1$ of $\mathbf{B}$ contain the least square solution vectors. The residual sum of squares * for each column is given by the sum of squares of elements $N$ to $M-1$ in that column. * * -# If(!transpose) and $M < N$: find minimum norm solutions of underdetermined systems - * $A * X = B$.\n - * Upon exit the columns of $B$ contain the minimum norm solution vectors. + * $\mathbf{A} \cdot \mathbf{X} = \mathbf{B}$.\n + * Upon exit the columns of $\mathbf{B}$ contain the minimum norm solution vectors. * * -# If(transpose) and $M \geq N$: find minimum norm solutions of underdetermined system - * $ A^\top X = B$.\n - * Upon exit the columns of $B$ contain the minimum norm solution vectors. + * $ \mathbf{A}^\top \cdot \mathbf{X} = \mathbf{B}$.\n + * Upon exit the columns of $\mathbf{B}$ contain the minimum norm solution vectors. * * -# If(transpose) and $M < N$: least squares solution of overdetermined system - * $\min \Vert B - A^\top X\Vert$.\n + * $\min \Vert \mathbf{B} - \mathbf{A}^\top \cdot \mathbf{X}\Vert$.\n * Upon exit the rows $0$ to $M-1$ contain the least square solution vectors. The residual sum of squares * for each column is given by the sum of squares of elements $M$ to $N-1$ in that column. * - * If(!tranpose) then $B \in \mathbb{R}^{M \times N_{\rm RHS}}$, - * otherwise $B \in \mathbb{R}^{N \times N_{\rm RHS}}}$. - * The matrices $A$ and $B$ must have an identical block cyclic distribution for rows and columns. + * If(!tranpose) then $\mathbf{B} \in \mathbb{R}^{M \times N_{\rm RHS}}$, + * otherwise $\mathbf{B} \in \mathbb{R}^{N \times N_{\rm RHS}}$. + * The matrices $\mathbf{A}$ and $\mathbf{B}$ must have an identical block cyclic distribution for rows and columns. */ void least_squares(ScaLAPACKMatrix &B, const bool transpose=false); @@ -491,7 +492,7 @@ public: * Cholesky factorization (see l1_norm()). * * @note An alternative is to compute the inverse of the matrix - * explicitly and manually construct $k_1 = ||A||_1 ||A^{-1}||_1$. + * explicitly and manually construct $k_1 = ||\mathbf{A}||_1 \, ||\mathbf{A}^{-1}||_1$. */ NumberType reciprocal_condition_number(const NumberType a_norm) const;