rdoc/eigen.rdoc in rb-gsl-1.16.0.2 vs rdoc/eigen.rdoc in rb-gsl-1.16.0.3.rc1

- old
+ new

@@ -1,18 +1,18 @@ # # = Eigensystems -# === {}[link:index.html"name="0.1] Contentes -# 1. {Modules and classes}[link:rdoc/eigen_rdoc.html#1] -# 1. {Real Symmetric Matrices}[link:rdoc/eigen_rdoc.html#2] -# 1. {Complex Hermitian Matrices}[link:rdoc/eigen_rdoc.html#3] -# 1. {Real Nonsymmetric Matrices}[link:rdoc/eigen_rdoc.html#4] (>= GSL-1.9) -# 1. {Real Generalized Symmetric-Definite Eigensystems}[link:rdoc/eigen_rdoc.html#5] (>= GSL-1.10) -# 1. {Complex Generalized Hermitian-Definite Eigensystems}[link:rdoc/eigen_rdoc.html#6] (>= GSL-1.10) -# 1. {Real Generalized Nonsymmetric Eigensystems}[link:rdoc/eigen_rdoc.html#7] (>= GSL-1.10) -# 1. {Sorting Eigenvalues and Eigenvectors }[link:rdoc/eigen_rdoc.html#8] +# === Contentes +# 1. {Modules and classes}[link:eigen_rdoc.html#label-Modules+and+classes] +# 1. {Real Symmetric Matrices}[link:eigen_rdoc.html#label-Real+Symmetric+Matrices] +# 1. {Complex Hermitian Matrices}[link:eigen_rdoc.html#label-Complex+Hermitian+Matrices] +# 1. {Real Nonsymmetric Matrices}[link:eigen_rdoc.html#label-Real+Nonsymmetric+Matrices+%28%3E%3D+GSL-1.9%29] (>= GSL-1.9) +# 1. {Real Generalized Symmetric-Definite Eigensystems}[link:eigen_rdoc.html#label-Real+Generalized+Symmetric-Definite+Eigensystems+%28GSL-1.10%29] (>= GSL-1.10) +# 1. {Complex Generalized Hermitian-Definite Eigensystems}[link:eigen_rdoc.html#label-Complex+Generalized+Hermitian-Definite+Eigensystems+%28%3E%3D+GSL-1.10%29] (>= GSL-1.10) +# 1. {Real Generalized Nonsymmetric Eigensystems}[link:eigen_rdoc.html#label-Real+Generalized+Nonsymmetric+Eigensystems+%28%3E%3D+GSL-1.10%29] (>= GSL-1.10) +# 1. {Sorting Eigenvalues and Eigenvectors }[link:eigen_rdoc.html#label-Sorting+Eigenvalues+and+Eigenvectors] # -# == {}[link:index.html"name="1] Modules and classes +# == Modules and classes # # * GSL # * Eigen # * EigenValues < Vector # * EigenVectors < Matrix @@ -40,37 +40,37 @@ # * Gen (Module, >= GSL-1.10) # * Workspace (Class) # * Genv (Module, >= GSL-1.10) # * Workspace (Class) # -# == {}[link:index.html"name="2] Real Symmetric Matrices, GSL::Eigen::Symm module -# === {}[link:index.html"name="2.1] Workspace classes +# == Real Symmetric Matrices +# === Workspace classes # --- # * GSL::Eigen::Symm::Workspace.alloc(n) # * GSL::Eigen::Symmv::Workspace.alloc(n) # * GSL::Eigen::Herm::Workspace.alloc(n) # * GSL::Eigen::Hermv::Workspace.alloc(n) # # -# === {}[link:index.html"name="2.2] Methods to solve eigensystems +# === Methods to solve eigensystems # --- # * GSL::Eigen::symm(A) # * GSL::Eigen::symm(A, workspace) # * GSL::Matrix#eigen_symm # * GSL::Matrix#eigen_symm(workspace) # -# These methods compute the eigenvalues of the real symmetric matrix. +# These methods compute the eigenvalues of the real symmetric matrix. # The workspace object <tt>workspace</tt> can be omitted. # # --- # * GSL::Eigen::symmv(A) # * GSL::Matrix#eigen_symmv # -# These methods compute the eigenvalues and eigenvectors of the real symmetric +# These methods compute the eigenvalues and eigenvectors of the real symmetric # matrix, and return an array of two elements: -# The first is a <tt>GSL::Vector</tt> object which stores all the eigenvalues. -# The second is a <tt>GSL::Matrix object</tt>, whose columns contain +# The first is a <tt>GSL::Vector</tt> object which stores all the eigenvalues. +# The second is a <tt>GSL::Matrix object</tt>, whose columns contain # eigenvectors. # # 1. Singleton method of the <tt>GSL::Eigen</tt> module, <tt>GSL::Eigen::symm</tt> # # m = GSL::Matrix.alloc([1.0, 1/2.0, 1/3.0, 1/4.0], [1/2.0, 1/3.0, 1/4.0, 1/5.0], @@ -79,88 +79,88 @@ # # 1. Instance method of <tt>GSL::Matrix</tt> class # # eigval, eigvec = m.eigen_symmv # -# == {}[link:index.html"name="3] Complex Hermitian Matrices +# == Complex Hermitian Matrices # --- # * GSL::Eigen::herm(A) # * GSL::Eigen::herm(A, workspace) # * GSL::Matrix::Complex#eigen_herm # * GSL::Matrix::Complex#eigen_herm(workspace) # -# These methods compute the eigenvalues of the complex hermitian matrix. +# These methods compute the eigenvalues of the complex hermitian matrix. # # --- # * GSL::Eigen::hermv(A) # * GSL::Eigen::hermv(A, workspace) # * GSL::Matrix::Complex#eigen_hermv # * GSL::Matrix::Complex#eigen_hermv(workspace # # -# == {}[link:index.html"name="4] Real Nonsymmetric Matrices (>= GSL-1.9) +# == Real Nonsymmetric Matrices (>= GSL-1.9) # # --- # * GSL::Eigen::Nonsymm.alloc(n) # -# This allocates a workspace for computing eigenvalues of n-by-n real +# This allocates a workspace for computing eigenvalues of n-by-n real # nonsymmetric matrices. The size of the workspace is O(2n). # # --- # * GSL::Eigen::Nonsymm::params(compute_t, balance, wspace) # * GSL::Eigen::Nonsymm::Workspace#params(compute_t, balance) # -# This method sets some parameters which determine how the eigenvalue +# This method sets some parameters which determine how the eigenvalue # problem is solved in subsequent calls to <tt>GSL::Eigen::nonsymm</tt>. -# If <tt>compute_t</tt> is set to 1, the full Schur form <tt>T</tt> will be -# computed by gsl_eigen_nonsymm. If it is set to 0, <tt>T</tt> will not be -# computed (this is the default setting). -# Computing the full Schur form <tt>T</tt> requires approximately 1.5-2 times +# If <tt>compute_t</tt> is set to 1, the full Schur form <tt>T</tt> will be +# computed by gsl_eigen_nonsymm. If it is set to 0, <tt>T</tt> will not be +# computed (this is the default setting). +# Computing the full Schur form <tt>T</tt> requires approximately 1.5-2 times # the number of flops. # -# If <tt>balance</tt> is set to 1, a balancing transformation is applied to -# the matrix prior to computing eigenvalues. This transformation is designed -# to make the rows and columns of the matrix have comparable norms, and can -# result in more accurate eigenvalues for matrices whose entries vary widely +# If <tt>balance</tt> is set to 1, a balancing transformation is applied to +# the matrix prior to computing eigenvalues. This transformation is designed +# to make the rows and columns of the matrix have comparable norms, and can +# result in more accurate eigenvalues for matrices whose entries vary widely # in magnitude. See section Balancing for more information. Note that the -# balancing transformation does not preserve the orthogonality of the Schur -# vectors, so if you wish to compute the Schur vectors with -# <tt>GSL::Eigen::nonsymm_Z</tt> you will obtain the Schur vectors of the -# balanced matrix instead of the original matrix. The relationship will be -# where Q is the matrix of Schur vectors for the balanced matrix, and <tt>D</tt> -# is the balancing transformation. Then <tt>GSL::Eigen::nonsymm_Z</tt> will -# compute a matrix <tt>Z</tt> which satisfies with <tt>Z = D Q</tt>. +# balancing transformation does not preserve the orthogonality of the Schur +# vectors, so if you wish to compute the Schur vectors with +# <tt>GSL::Eigen::nonsymm_Z</tt> you will obtain the Schur vectors of the +# balanced matrix instead of the original matrix. The relationship will be +# where Q is the matrix of Schur vectors for the balanced matrix, and <tt>D</tt> +# is the balancing transformation. Then <tt>GSL::Eigen::nonsymm_Z</tt> will +# compute a matrix <tt>Z</tt> which satisfies with <tt>Z = D Q</tt>. # Note that <tt>Z</tt> will not be orthogonal. For this reason, balancing is # not performed by default. # # --- # * GSL::Eigen::nonsymm(m, eval, wspace) # * GSL::Eigen::nonsymm(m) # * GSL::Matrix#eigen_nonsymm() # * GSL::Matrix#eigen_nonsymm(wspace) # * GSL::Matrix#eigen_nonsymm(eval, wspace) # -# These methods compute the eigenvalues of the real nonsymmetric matrix <tt>m</tt> -# and return them, or store in the vector <tt>eval</tt> if it given. -# If <tt>T</tt> is desired, it is stored in <tt>m</tt> on output, however the lower -# triangular portion will not be zeroed out. Otherwise, on output, the diagonal -# of <tt>m</tt> will contain the 1-by-1 real eigenvalues and 2-by-2 complex -# conjugate eigenvalue systems, and the rest of <tt>m</tt> is destroyed. +# These methods compute the eigenvalues of the real nonsymmetric matrix <tt>m</tt> +# and return them, or store in the vector <tt>eval</tt> if it given. +# If <tt>T</tt> is desired, it is stored in <tt>m</tt> on output, however the lower +# triangular portion will not be zeroed out. Otherwise, on output, the diagonal +# of <tt>m</tt> will contain the 1-by-1 real eigenvalues and 2-by-2 complex +# conjugate eigenvalue systems, and the rest of <tt>m</tt> is destroyed. # # --- # * GSL::Eigen::nonsymm_Z(m, eval, Z, wspace) # * GSL::Eigen::nonsymm_Z(m) # * GSL::Matrix#eigen_nonsymm_Z() # * GSL::Matrix#eigen_nonsymm(eval, Z, wspace) # -# These methods are identical to <tt>GSL::Eigen::nonsymm</tt> except they also +# These methods are identical to <tt>GSL::Eigen::nonsymm</tt> except they also # compute the Schur vectors and return them (or store into <tt>Z</tt>). # # --- # * GSL::Eigen::Nonsymmv.alloc(n) # -# Allocates a workspace for computing eigenvalues and eigenvectors +# Allocates a workspace for computing eigenvalues and eigenvectors # of n-by-n real nonsymmetric matrices. The size of the workspace is O(5n). # --- # * GSL::Eigen::nonsymm(m) # * GSL::Eigen::nonsymm(m, wspace) # * GSL::Eigen::nonsymm(m, eval, evec) @@ -168,85 +168,85 @@ # * GSL::Matrix#eigen_nonsymmv() # * GSL::Matrix#eigen_nonsymmv(wspace) # * GSL::Matrix#eigen_nonsymmv(eval, evec) # * GSL::Matrix#eigen_nonsymmv(eval, evec, wspace) # -# Compute eigenvalues and right eigenvectors of the n-by-n real nonsymmetric +# Compute eigenvalues and right eigenvectors of the n-by-n real nonsymmetric # matrix. The computed eigenvectors are normalized to have Euclidean norm 1. -# On output, the upper portion of <tt>m</tt> contains the Schur form <tt>T</tt>. +# On output, the upper portion of <tt>m</tt> contains the Schur form <tt>T</tt>. # -# == {}[link:index.html"name="5] Real Generalized Symmetric-Definite Eigensystems (GSL-1.10) -# The real generalized symmetric-definite eigenvalue problem is to -# find eigenvalues <tt>lambda</tt> and eigenvectors <tt>x</tt> such that -# where <tt>A</tt> and <tt>B</tt> are symmetric matrices, and <tt>B</tt> +# == Real Generalized Symmetric-Definite Eigensystems (GSL-1.10) +# The real generalized symmetric-definite eigenvalue problem is to +# find eigenvalues <tt>lambda</tt> and eigenvectors <tt>x</tt> such that +# where <tt>A</tt> and <tt>B</tt> are symmetric matrices, and <tt>B</tt> # is positive-definite. This problem reduces to the standard symmetric eigenvalue -# problem by applying the Cholesky decomposition to <tt>B</tt>: -# Therefore, the problem becomes <tt>C y = lambda y</tt> -# where <tt>C = L^{-1} A L^{-t}</tt> is symmetric, and <tt>y = L^t x</tt>. -# The standard symmetric eigensolver can be applied to the matrix <tt>C</tt>. -# The resulting eigenvectors are backtransformed to find the vectors of the -# original problem. The eigenvalues and eigenvectors of the generalized -# symmetric-definite eigenproblem are always real. +# problem by applying the Cholesky decomposition to <tt>B</tt>: +# Therefore, the problem becomes <tt>C y = lambda y</tt> +# where <tt>C = L^{-1} A L^{-t}</tt> is symmetric, and <tt>y = L^t x</tt>. +# The standard symmetric eigensolver can be applied to the matrix <tt>C</tt>. +# The resulting eigenvectors are backtransformed to find the vectors of the +# original problem. The eigenvalues and eigenvectors of the generalized +# symmetric-definite eigenproblem are always real. # # --- # * GSL::Eigen::Gensymm.alloc(n) # * GSL::Eigen::Gensymm::Workspace.alloc(n) # -# Allocates a workspace for computing eigenvalues of n-by-n real -# generalized symmetric-definite eigensystems. -# The size of the workspace is O(2n). +# Allocates a workspace for computing eigenvalues of n-by-n real +# generalized symmetric-definite eigensystems. +# The size of the workspace is O(2n). # --- # * GSL::Eigen::gensymm(A, B, w) # -# Computes the eigenvalues of the real generalized symmetric-definite matrix -# pair <tt>A, B</tt>, and returns them as a <tt>GSL::Vector</tt>, +# Computes the eigenvalues of the real generalized symmetric-definite matrix +# pair <tt>A, B</tt>, and returns them as a <tt>GSL::Vector</tt>, # using the method outlined above. On output, B contains its Cholesky # decomposition. # --- # * GSL::Eigen::gensymmv(A, B, w) # -# Computes the eigenvalues and eigenvectors of the real generalized -# symmetric-definite matrix pair <tt>A, B</tt>, and returns -# them as a <tt>GSL::Vector</tt> and a <tt>GSL::Matrix</tt>. -# The computed eigenvectors are normalized to have unit magnitude. +# Computes the eigenvalues and eigenvectors of the real generalized +# symmetric-definite matrix pair <tt>A, B</tt>, and returns +# them as a <tt>GSL::Vector</tt> and a <tt>GSL::Matrix</tt>. +# The computed eigenvectors are normalized to have unit magnitude. # On output, <tt>B</tt> contains its Cholesky decomposition. # -# == {}[link:index.html"name="6] Complex Generalized Hermitian-Definite Eigensystems (>= GSL-1.10) -# The complex generalized hermitian-definite eigenvalue problem is to -# find eigenvalues <tt>lambda</tt> and eigenvectors <tt>x</tt> such that -# where <tt>A</tt> and <tt>B</tt> are hermitian matrices, and <tt>B</tt> -# is positive-definite. Similarly to the real case, this can be reduced to -# <tt>C y = lambda y</tt> where <tt>C = L^{-1} A L^{-H}</tt> is hermitian, -# and <tt>y = L^H x</tt>. The standard hermitian eigensolver can be applied to -# the matrix <tt>C</tt>. The resulting eigenvectors are backtransformed -# to find the vectors of the original problem. -# The eigenvalues of the generalized hermitian-definite eigenproblem are always -# real. +# == Complex Generalized Hermitian-Definite Eigensystems (>= GSL-1.10) +# The complex generalized hermitian-definite eigenvalue problem is to +# find eigenvalues <tt>lambda</tt> and eigenvectors <tt>x</tt> such that +# where <tt>A</tt> and <tt>B</tt> are hermitian matrices, and <tt>B</tt> +# is positive-definite. Similarly to the real case, this can be reduced to +# <tt>C y = lambda y</tt> where <tt>C = L^{-1} A L^{-H}</tt> is hermitian, +# and <tt>y = L^H x</tt>. The standard hermitian eigensolver can be applied to +# the matrix <tt>C</tt>. The resulting eigenvectors are backtransformed +# to find the vectors of the original problem. +# The eigenvalues of the generalized hermitian-definite eigenproblem are always +# real. # # --- # * GSL::Eigen::Genherm.alloc(n) # -# Allocates a workspace for computing eigenvalues of n-by-n complex -# generalized hermitian-definite eigensystems. -# The size of the workspace is O(3n). +# Allocates a workspace for computing eigenvalues of n-by-n complex +# generalized hermitian-definite eigensystems. +# The size of the workspace is O(3n). # --- # * GSL::Eigen::genherm(A, B, w) # -# Computes the eigenvalues of the complex generalized hermitian-definite -# matrix pair <tt>A, B</tt>, and returns them as a <tt>GSL::Vector</tt>, -# using the method outlined above. +# Computes the eigenvalues of the complex generalized hermitian-definite +# matrix pair <tt>A, B</tt>, and returns them as a <tt>GSL::Vector</tt>, +# using the method outlined above. # On output, <tt>B</tt> contains its Cholesky decomposition. # --- # * GSL::Eigen::genherm(A, B, w) # -# Computes the eigenvalues and eigenvectors of the complex generalized -# hermitian-definite matrix pair <tt>A, B</tt>, -# and returns them as a <tt>GSL::Vector</tt> and a <tt>GSL::Matrix::Complex</tt>. -# The computed eigenvectors are normalized to have unit magnitude. +# Computes the eigenvalues and eigenvectors of the complex generalized +# hermitian-definite matrix pair <tt>A, B</tt>, +# and returns them as a <tt>GSL::Vector</tt> and a <tt>GSL::Matrix::Complex</tt>. +# The computed eigenvectors are normalized to have unit magnitude. # On output, <tt>B</tt> contains its Cholesky decomposition. # -# == {}[link:index.html"name="7] Real Generalized Nonsymmetric Eigensystems (>= GSL-1.10) +# == Real Generalized Nonsymmetric Eigensystems (>= GSL-1.10) # # --- # * GSL::Eigen::Gen.alloc(n) # * GSL::Eigen::Gen::Workspace.alloc(n) # @@ -255,83 +255,83 @@ # # --- # * GSL::Eigen::Gen::params(compute_s, compute_t, balance, w) # * GSL::Eigen::gen_params(compute_s, compute_t, balance, w) # -# Set some parameters which determine how the eigenvalue problem is solved +# Set some parameters which determine how the eigenvalue problem is solved # in subsequent calls to <tt>GSL::Eigen::gen</tt>. # -# If <tt>compute_s</tt> is set to 1, the full Schur form <tt>S</tt> will be -# computed by <tt>GSL::Eigen::gen</tt>. If it is set to 0, <tt>S</tt> will -# not be computed (this is the default setting). <tt>S</tt> is a quasi upper -# triangular matrix with 1-by-1 and 2-by-2 blocks on its diagonal. -# 1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks -# correspond to complex eigenvalues. +# If <tt>compute_s</tt> is set to 1, the full Schur form <tt>S</tt> will be +# computed by <tt>GSL::Eigen::gen</tt>. If it is set to 0, <tt>S</tt> will +# not be computed (this is the default setting). <tt>S</tt> is a quasi upper +# triangular matrix with 1-by-1 and 2-by-2 blocks on its diagonal. +# 1-by-1 blocks correspond to real eigenvalues, and 2-by-2 blocks +# correspond to complex eigenvalues. # -# If <tt>compute_t</tt> is set to 1, the full Schur form <tt>T</tt> will -# be computed by <tt>GSL::Eigen::gen</tt>. If it is set to 0, <tt>T</tt> -# will not be computed (this is the default setting). <tt>T</tt> -# is an upper triangular matrix with non-negative elements on its diagonal. -# Any 2-by-2 blocks in <tt>S</tt> will correspond to a 2-by-2 diagonal block -# in <tt>T</tt>. +# If <tt>compute_t</tt> is set to 1, the full Schur form <tt>T</tt> will +# be computed by <tt>GSL::Eigen::gen</tt>. If it is set to 0, <tt>T</tt> +# will not be computed (this is the default setting). <tt>T</tt> +# is an upper triangular matrix with non-negative elements on its diagonal. +# Any 2-by-2 blocks in <tt>S</tt> will correspond to a 2-by-2 diagonal block +# in <tt>T</tt>. # -# The <tt>balance</tt> parameter is currently ignored, since generalized -# balancing is not yet implemented. +# The <tt>balance</tt> parameter is currently ignored, since generalized +# balancing is not yet implemented. # # --- # * GSL::Eigen::gen(A, B, w) # # Computes the eigenvalues of the real generalized nonsymmetric matrix pair -# <tt>A, B</tt>, and returns them as pairs in (alpha, beta), -# where alpha is <tt>GSL::Vector::Complex</tt> and beta is <tt>GSL::Vector</tt>. +# <tt>A, B</tt>, and returns them as pairs in (alpha, beta), +# where alpha is <tt>GSL::Vector::Complex</tt> and beta is <tt>GSL::Vector</tt>. # If beta_i is non-zero, then lambda = alpha_i / beta_i is an eigenvalue. -# Likewise, if alpha_i is non-zero, then mu = beta_i / alpha_i is an -# eigenvalue of the alternate problem mu A y = B y. -# The elements of <tt>beta</tt> are normalized to be non-negative. +# Likewise, if alpha_i is non-zero, then mu = beta_i / alpha_i is an +# eigenvalue of the alternate problem mu A y = B y. +# The elements of <tt>beta</tt> are normalized to be non-negative. # -# If <tt>S</tt> is desired, it is stored in <tt>A</tt> on output. -# If <tt>T</tt> is desired, it is stored in <tt>B</tt> on output. -# The ordering of eigenvalues in <tt>alpha, beta</tt> +# If <tt>S</tt> is desired, it is stored in <tt>A</tt> on output. +# If <tt>T</tt> is desired, it is stored in <tt>B</tt> on output. +# The ordering of eigenvalues in <tt>alpha, beta</tt> # follows the ordering of the diagonal blocks in the Schur forms <tt>S</tt> -# and <tt>T</tt>. +# and <tt>T</tt>. # # --- # * GSL::Eigen::gen_QZ(A, B, w) # -# This method is identical to <tt>GSL::Eigen::gen</tt> except it also computes +# This method is identical to <tt>GSL::Eigen::gen</tt> except it also computes # the left and right Schur vectors and returns them. # # --- # * GSL::Eigen::Genv.alloc(n) # * GSL::Eigen::Genv::Workspace.alloc(n) # -# Allocatesa workspace for computing eigenvalues and eigenvectors of -# n-by-n real generalized nonsymmetric eigensystems. -# The size of the workspace is O(7n). +# Allocatesa workspace for computing eigenvalues and eigenvectors of +# n-by-n real generalized nonsymmetric eigensystems. +# The size of the workspace is O(7n). # # --- # * GSL::Eigen::genv(A, B, w) # -# Computes eigenvalues and right eigenvectors of the n-by-n real generalized -# nonsymmetric matrix pair <tt>A, B</tt>. The eigenvalues and eigenvectors -# are returned in <tt>alpha, beta, evec</tt>. -# On output, <tt>A, B</tt> contains the generalized Schur form <tt>S, T</tt>. +# Computes eigenvalues and right eigenvectors of the n-by-n real generalized +# nonsymmetric matrix pair <tt>A, B</tt>. The eigenvalues and eigenvectors +# are returned in <tt>alpha, beta, evec</tt>. +# On output, <tt>A, B</tt> contains the generalized Schur form <tt>S, T</tt>. # # --- # * GSL::Eigen::genv_QZ(A, B, w) # -# This method is identical to <tt>GSL::Eigen::genv</tt> except it also computes +# This method is identical to <tt>GSL::Eigen::genv</tt> except it also computes # the left and right Schur vectors and returns them. # -# == {}[link:index.html"name="8] Sorting Eigenvalues and Eigenvectors +# == Sorting Eigenvalues and Eigenvectors # --- # * GSL::Eigen::symmv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # * GSL::Eigen::Symmv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # -# These methods simultaneously sort the eigenvalues stored in the vector -# <tt>eval</tt> and the corresponding real eigenvectors stored in the -# columns of the matrix <tt>evec</tt> into ascending or descending order +# These methods simultaneously sort the eigenvalues stored in the vector +# <tt>eval</tt> and the corresponding real eigenvectors stored in the +# columns of the matrix <tt>evec</tt> into ascending or descending order # according to the value of the parameter <tt>type</tt>, # # * <tt>GSL::Eigen::SORT_VAL_ASC</tt> # ascending order in numerical value # * <tt>GSL::Eigen::SORT_VAL_DESC</tt> @@ -345,57 +345,57 @@ # # --- # * GSL::Eigen::hermv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # * GSL::Eigen::Hermv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # -# These methods simultaneously sort the eigenvalues stored in the vector -# <tt>eval</tt> and the corresponding complex eigenvectors stored in the columns -# of the matrix <tt>evec</tt> into ascending or descending order according +# These methods simultaneously sort the eigenvalues stored in the vector +# <tt>eval</tt> and the corresponding complex eigenvectors stored in the columns +# of the matrix <tt>evec</tt> into ascending or descending order according # to the value of the parameter <tt>type</tt> as shown above. # # --- # * GSL::Eigen::nonsymmv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # * GSL::Eigen::Nonsymmv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # -# Sorts the eigenvalues stored in the vector <tt>eval</tt> and the corresponding -# complex eigenvectors stored in the columns of the matrix <tt>evec</tt> -# into ascending or descending order according to the value of the -# parameter <tt>type</tt> as shown above. -# Only <tt>GSL::EIGEN_SORT_ABS_ASC</tt> and <tt>GSL::EIGEN_SORT_ABS_DESC</tt> -# are supported due to the eigenvalues being complex. +# Sorts the eigenvalues stored in the vector <tt>eval</tt> and the corresponding +# complex eigenvectors stored in the columns of the matrix <tt>evec</tt> +# into ascending or descending order according to the value of the +# parameter <tt>type</tt> as shown above. +# Only <tt>GSL::EIGEN_SORT_ABS_ASC</tt> and <tt>GSL::EIGEN_SORT_ABS_DESC</tt> +# are supported due to the eigenvalues being complex. # # --- # * GSL::Eigen::gensymmv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # * GSL::Eigen::Gensymmv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # -# Sorts the eigenvalues stored in the vector <tt>eval</tt> and the -# corresponding real eigenvectors stored in the columns of the matrix -# <tt>evec</tt> into ascending or descending order according to the value of -# the parameter <tt>type</tt> as shown above. +# Sorts the eigenvalues stored in the vector <tt>eval</tt> and the +# corresponding real eigenvectors stored in the columns of the matrix +# <tt>evec</tt> into ascending or descending order according to the value of +# the parameter <tt>type</tt> as shown above. # # --- # * GSL::Eigen::gensymmv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # * GSL::Eigen::Gensymmv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC) # -# Sorts the eigenvalues stored in the vector <tt>eval</tt> and the -# corresponding complex eigenvectors stored in the columns of the matrix -# <tt>evec</tt> into ascending or descending order according to the value of -# the parameter <tt>type</tt> as shown above. +# Sorts the eigenvalues stored in the vector <tt>eval</tt> and the +# corresponding complex eigenvectors stored in the columns of the matrix +# <tt>evec</tt> into ascending or descending order according to the value of +# the parameter <tt>type</tt> as shown above. # # --- # * GSL::Eigen::genv_sort(alpha, beta, evec, type = GSL::Eigen::SORT_VAL_ASC) # * GSL::Eigen::Genv::sort(alpha, beta, evec, type = GSL::Eigen::SORT_VAL_ASC) # -# Sorts the eigenvalues stored in the vectors <tt>alpha, beta</tt> and the -# corresponding complex eigenvectors stored in the columns of the matrix -# <tt>evec</tt> into ascending or descending order according to the value of +# Sorts the eigenvalues stored in the vectors <tt>alpha, beta</tt> and the +# corresponding complex eigenvectors stored in the columns of the matrix +# <tt>evec</tt> into ascending or descending order according to the value of # the parameter <tt>type</tt> as shown above. Only <tt>GSL::EIGEN_SORT_ABS_ASC</tt> -# and <tt>GSL::EIGEN_SORT_ABS_DESC</tt> are supported due to the eigenvalues -# being complex. +# and <tt>GSL::EIGEN_SORT_ABS_DESC</tt> are supported due to the eigenvalues +# being complex. # -# {prev}[link:rdoc/linalg_rdoc.html] -# {next}[link:rdoc/fft_rdoc.html] +# {prev}[link:linalg_rdoc.html] +# {next}[link:fft_rdoc.html] # -# {Reference index}[link:rdoc/ref_rdoc.html] +# {Reference index}[link:ref_rdoc.html] # {top}[link:index.html] # #