Condition Number Estimation of Preconditioned Matrices
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{"title"=>"Condition number estimation of preconditioned matrices", "type"=>"journal", "authors"=>[{"first_name"=>"Noriyuki", "last_name"=>"Kushida", "scopus_author_id"=>"23012086700"}], "year"=>2015, "source"=>"PLoS ONE", "identifiers"=>{"pui"=>"604114879", "issn"=>"19326203", "doi"=>"10.1371/journal.pone.0122331", "sgr"=>"84929484926", "scopus"=>"2-s2.0-84929484926"}, "id"=>"f7dc939a-945e-30a2-ad83-8111c0e97a02", "link"=>"http://www.mendeley.com/research/condition-number-estimation-preconditioned-matrices", "reader_count"=>2, "reader_count_by_academic_status"=>{"Student > Ph. D. Student"=>2}, "reader_count_by_user_role"=>{"Student > Ph. D. Student"=>2}, "reader_count_by_subject_area"=>{"Mathematics"=>1, "Computer Science"=>1}, "reader_count_by_subdiscipline"=>{"Computer Science"=>{"Computer Science"=>1}, "Mathematics"=>{"Mathematics"=>1}}, "group_count"=>0}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1992470"], "description"=>"<p>In the table, the eigenvalues of a 10 × 10 diagonal matrix calculated using the Lanczos connection method are listed. The first row indicates the eigenvalues with all information, and the second row indicates the eigenvalues without the first CG step information. In the full information case, the condition number becomes unreasonable, while the second one shows a reasonable result. In the table, NA means “Not Available”.</p><p>Eigenvalues of a 10 × 10 diagonal matrix calculated using the Lanczos connection method.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358816, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t006", "stats"=>{"downloads"=>6, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Eigenvalues_of_a_10_215_10_diagonal_matrix_calculated_using_the_Lanczos_connection_method_/1358816", "title"=>"Eigenvalues of a 10 × 10 diagonal matrix calculated using the Lanczos connection method.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992468"], "description"=>"<p>Pseudocode of the PCG in the modified Hager’s method. In the algorithm,⟨⋅, ⋅⟩ denotes vector inner product.</p><p>Preconditioned Conjugte Gradient Algorithm in the modified Hager’s method.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358814, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t004", "stats"=>{"downloads"=>10, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Preconditioned_Conjugte_Gradient_Algorithm_in_the_modified_Hager_8217_s_method_/1358814", "title"=>"Preconditioned Conjugte Gradient Algorithm in the modified Hager’s method.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992465"], "description"=>"<p>Pseudocode of the PCG. In the algorithm,⟨⋅, ⋅⟩ denotes vector inner product.</p><p>Preconditioned Conjugte Gradient Algorithm.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358811, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t001", "stats"=>{"downloads"=>3, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Preconditioned_Conjugte_Gradient_Algorithm_/1358811", "title"=>"Preconditioned Conjugte Gradient Algorithm.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992476"], "description"=>"<p>The condition numbers of SSOR preconditioned Poisson–FEM matrices using the Lanczos connection method, the modified Hager’s method, and Octave are listed. In this table, error values defined as </p><p></p><p><mi>E</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo></p><p></p><p><mo stretchy=\"true\">∣</mo>Calculatedconditionnumber<mo>−</mo>Octave<mo stretchy=\"true\">∣</mo></p>Octave<p></p><mo>×</mo><mn>100</mn><mo stretchy=\"false\">(</mo><mo>%</mo><mo stretchy=\"false\">)</mo><p></p><p></p> are also indicated.<p></p><p>Condition numbers: SSOR preconditioned Poisson–FEM matrix.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358822, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t012", "stats"=>{"downloads"=>4, "page_views"=>17, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Condition_numbers_SSOR_preconditioned_Poisson_8211_FEM_matrix_/1358822", "title"=>"Condition numbers: SSOR preconditioned Poisson–FEM matrix.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992473"], "description"=>"<p>The condition numbers of diagonal scaled <b>T</b><sub><i>Tri</i></sub> using the Lanczos connection method, the modified Hager’s method, and Octave are listed.</p><p>Condition numbers: Diagonal scaled tridiagonal matrix.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358819, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t009", "stats"=>{"downloads"=>2, "page_views"=>9, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Condition_numbers_Diagonal_scaled_tridiagonal_matrix_/1358819", "title"=>"Condition numbers: Diagonal scaled tridiagonal matrix.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992472"], "description"=>"<p>The condition numbers of <b>T</b><sub><i>Tri</i></sub> using the Lanczos connection method, the modified Hager’s method, and Octave are listed.</p><p>Condition numbers: Tridiagonal matrix.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358818, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t008", "stats"=>{"downloads"=>4, "page_views"=>15, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Condition_numbers_Tridiagonal_matrix_/1358818", "title"=>"Condition numbers: Tridiagonal matrix.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992467"], "description"=>"<p>Pseudocode of the modified Hager’s method. The modified Hager’s method consists of two major parts: one is for calculating </p><p></p><p><mo stretchy=\"false\">‖</mo></p><p></p><p><mo stretchy=\"true\">(</mo></p><p>M<mn>1</mn></p><p><mo>−</mo><mn>1</mn></p><p></p><p>AM<mn>2</mn></p><p><mo>−</mo><mn>1</mn></p><p></p><mo stretchy=\"true\">)</mo><p></p><p><mo>−</mo><mn>1</mn></p><p></p><p><mo stretchy=\"false\">‖</mo><mn>1</mn></p><p></p><p></p> and the other is for <p></p><p><mo stretchy=\"false\">‖</mo></p><p>M<mn>1</mn></p><p><mo>−</mo><mn>1</mn></p><p></p><p>AM<mn>2</mn></p><p><mo>−</mo><mn>1</mn></p><p></p><p><mo stretchy=\"false\">‖</mo><mn>1</mn></p><p></p><p></p>. In the algorithm, [<i>i</i>] denotes the <i>i</i>th vector element of a vector.<p></p><p>The algorithm of the modified Hager’s method.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358813, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t003", "stats"=>{"downloads"=>5, "page_views"=>14, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_algorithm_of_the_modified_Hager_8217_s_method_/1358813", "title"=>"The algorithm of the modified Hager’s method.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992466"], "description"=>"<p>Pseudocode of Hager’s method. In the algorithm, [<i>i</i>] denotes the <i>i</i>th vector element of a vector.</p><p>The algorithm of Hager’s method.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358812, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t002", "stats"=>{"downloads"=>5, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_algorithm_of_Hager_8217_s_method_/1358812", "title"=>"The algorithm of Hager’s method.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992475"], "description"=>"<p>The condition numbers of the diagonal scaled Poisson–FEM matrices using the Lanczos connection method, the modified Hager’s method, and Octave are listed. In this table, error values defined as </p><p></p><p><mi>E</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo></p><p></p><p><mo stretchy=\"true\">∣</mo>Calculatedconditionnumber<mo>−</mo>Octave<mo stretchy=\"true\">∣</mo></p>Octave<p></p><mo>×</mo><mn>100</mn><mo stretchy=\"false\">(</mo><mo>%</mo><mo stretchy=\"false\">)</mo><p></p><p></p> are also indicated.<p></p><p>Condition numbers: Diagonal scaled Poisson–FEM matrix.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358821, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t011", "stats"=>{"downloads"=>3, "page_views"=>23, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Condition_numbers_Diagonal_scaled_Poisson_8211_FEM_matrix_/1358821", "title"=>"Condition numbers: Diagonal scaled Poisson–FEM matrix.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992474"], "description"=>"<p>The condition numbers of SSOR preconditioned <b>T</b><sub><i>Pei</i></sub> using the Lanczos connection method, the modified Hager’s method, and Octave are listed. In this table, error values defined as </p><p></p><p><mi>E</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>=</mo></p><p></p><p><mo stretchy=\"true\">∣</mo>Calculatedconditionnumber<mo>−</mo>Octave<mo stretchy=\"true\">∣</mo></p>Octave<p></p><mo>×</mo><mn>100</mn><mo stretchy=\"false\">(</mo><mo>%</mo><mo stretchy=\"false\">)</mo><p></p><p></p> are also indicated.<p></p><p>Condition numbers: SSOR preconditioned Pei’s matrix for various <i>d</i>.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358820, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t010", "stats"=>{"downloads"=>4, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Condition_numbers_SSOR_preconditioned_Pei_s_matrix_for_various_d_/1358820", "title"=>"Condition numbers: SSOR preconditioned Pei’s matrix for various <i>d</i>.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992471"], "description"=>"<p>In the table, condition numbers calculated using both the Lanczos connection method and Hager’s method are listed with numbers with Octave. Both method show good performances, but the Lanczos method includes some error, although the test matrix is simple.</p><p>Condition numbers of <b>T</b><sub><i>diag</i></sub> by the Lanczos connection method and Hager’s method.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358817, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t007", "stats"=>{"downloads"=>4, "page_views"=>15, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Condition_numbers_of_T_diag_by_the_Lanczos_connection_method_and_Hager_s_method_/1358817", "title"=>"Condition numbers of <b>T</b><sub><i>diag</i></sub> by the Lanczos connection method and Hager’s method.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}
  • {"files"=>["https://ndownloader.figshare.com/files/1992469"], "description"=>"<p>Details of the computational environment used for the precision verification.</p><p>Computational Environment.</p>", "links"=>[], "tags"=>["Lanczos connection method", "matrix", "condition number", "SSOR", "matrice", "condition number estimation method", "Lanczos connection", "conjugate gradient method", "Condition Number Estimation"], "article_id"=>1358815, "categories"=>["Uncategorised"], "users"=>["Noriyuki Kushida"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0122331.t005", "stats"=>{"downloads"=>4, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Computational_Environment_/1358815", "title"=>"Computational Environment.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-27 03:49:51"}

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{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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