Three methods for solving systems of linear equations: Comparing the advantages and disadvantages Article Swipe
Haoyu Luo
,
Simeng Wu
,
Ningyun Xie
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.1088/1742-6596/2012/1/012061
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.1088/1742-6596/2012/1/012061
This paper aims to introduce some prevalent techniques which have been used to solve linear systems. Firstly, we introduce the simplest method: how to eliminate variables in which the original systems would not be changed. We then introduce Gaussian eliminations, which work on the augmented matrix derived from a linear system. To the end, we present Cramer’s rule, which computes the solution to a linear system based on matrix determinants. These methods have their application scenarios. For instance, eliminating variables has no limited condition for its operations. However, Cramer’s rule needs to be under the condition of square matrices. At the same time, Gaussian elimination requires three elementary row operations, and Gaussian elimination paves the way for computing the rank of matrices. The choice between these methods depends on coefficient matrices in terms of dimensions and matrix properties such as singularity.
Related Topics
Concepts
Gaussian elimination
Coefficient matrix
Gaussian
Rank (graph theory)
Matrix (chemical analysis)
Applied mathematics
Linear system
Computer science
System of linear equations
Augmented matrix
Singularity
Mathematical optimization
Mathematics
Algorithm
State-transition matrix
Symmetric matrix
Mathematical analysis
Physics
Composite material
Eigenvalues and eigenvectors
Combinatorics
Materials science
Quantum mechanics
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1088/1742-6596/2012/1/012061
- https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdf
- OA Status
- diamond
- Cited By
- 2
- References
- 1
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W3196753368
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W3196753368Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1088/1742-6596/2012/1/012061Digital Object Identifier
- Title
-
Three methods for solving systems of linear equations: Comparing the advantages and disadvantagesWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2021Year of publication
- Publication date
-
2021-09-01Full publication date if available
- Authors
-
Haoyu Luo, Simeng Wu, Ningyun XieList of authors in order
- Landing page
-
https://doi.org/10.1088/1742-6596/2012/1/012061Publisher landing page
- PDF URL
-
https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdfDirect link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
diamondOpen access status per OpenAlex
- OA URL
-
https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdfDirect OA link when available
- Concepts
-
Gaussian elimination, Coefficient matrix, Gaussian, Rank (graph theory), Matrix (chemical analysis), Applied mathematics, Linear system, Computer science, System of linear equations, Augmented matrix, Singularity, Mathematical optimization, Mathematics, Algorithm, State-transition matrix, Symmetric matrix, Mathematical analysis, Physics, Composite material, Eigenvalues and eigenvectors, Combinatorics, Materials science, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
2Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 1, 2024: 1Per-year citation counts (last 5 years)
- References (count)
-
1Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W3196753368 |
|---|---|
| doi | https://doi.org/10.1088/1742-6596/2012/1/012061 |
| ids.doi | https://doi.org/10.1088/1742-6596/2012/1/012061 |
| ids.mag | 3196753368 |
| ids.openalex | https://openalex.org/W3196753368 |
| fwci | 0.16009212 |
| type | article |
| title | Three methods for solving systems of linear equations: Comparing the advantages and disadvantages |
| biblio.issue | 1 |
| biblio.volume | 2012 |
| biblio.last_page | 012061 |
| biblio.first_page | 012061 |
| topics[0].id | https://openalex.org/T10792 |
| topics[0].field.id | https://openalex.org/fields/17 |
| topics[0].field.display_name | Computer Science |
| topics[0].score | 0.8367999792098999 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/1703 |
| topics[0].subfield.display_name | Computational Theory and Mathematics |
| topics[0].display_name | Matrix Theory and Algorithms |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C126312332 |
| concepts[0].level | 3 |
| concepts[0].score | 0.8746613264083862 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q2658 |
| concepts[0].display_name | Gaussian elimination |
| concepts[1].id | https://openalex.org/C60866291 |
| concepts[1].level | 3 |
| concepts[1].score | 0.7010838985443115 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q5140577 |
| concepts[1].display_name | Coefficient matrix |
| concepts[2].id | https://openalex.org/C163716315 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6544263958930969 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q901177 |
| concepts[2].display_name | Gaussian |
| concepts[3].id | https://openalex.org/C164226766 |
| concepts[3].level | 2 |
| concepts[3].score | 0.6495974659919739 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q7293202 |
| concepts[3].display_name | Rank (graph theory) |
| concepts[4].id | https://openalex.org/C106487976 |
| concepts[4].level | 2 |
| concepts[4].score | 0.5831514000892639 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q685816 |
| concepts[4].display_name | Matrix (chemical analysis) |
| concepts[5].id | https://openalex.org/C28826006 |
| concepts[5].level | 1 |
| concepts[5].score | 0.5599634051322937 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q33521 |
| concepts[5].display_name | Applied mathematics |
| concepts[6].id | https://openalex.org/C6802819 |
| concepts[6].level | 2 |
| concepts[6].score | 0.5407045483589172 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q1072174 |
| concepts[6].display_name | Linear system |
| concepts[7].id | https://openalex.org/C41008148 |
| concepts[7].level | 0 |
| concepts[7].score | 0.5079692006111145 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[7].display_name | Computer science |
| concepts[8].id | https://openalex.org/C94523830 |
| concepts[8].level | 2 |
| concepts[8].score | 0.48877617716789246 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q11203 |
| concepts[8].display_name | System of linear equations |
| concepts[9].id | https://openalex.org/C105129939 |
| concepts[9].level | 5 |
| concepts[9].score | 0.4852454662322998 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q2750301 |
| concepts[9].display_name | Augmented matrix |
| concepts[10].id | https://openalex.org/C16171025 |
| concepts[10].level | 2 |
| concepts[10].score | 0.45986106991767883 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q863349 |
| concepts[10].display_name | Singularity |
| concepts[11].id | https://openalex.org/C126255220 |
| concepts[11].level | 1 |
| concepts[11].score | 0.435200959444046 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q141495 |
| concepts[11].display_name | Mathematical optimization |
| concepts[12].id | https://openalex.org/C33923547 |
| concepts[12].level | 0 |
| concepts[12].score | 0.41063418984413147 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[12].display_name | Mathematics |
| concepts[13].id | https://openalex.org/C11413529 |
| concepts[13].level | 1 |
| concepts[13].score | 0.3379117548465729 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q8366 |
| concepts[13].display_name | Algorithm |
| concepts[14].id | https://openalex.org/C22629506 |
| concepts[14].level | 4 |
| concepts[14].score | 0.32817381620407104 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q7602993 |
| concepts[14].display_name | State-transition matrix |
| concepts[15].id | https://openalex.org/C54848796 |
| concepts[15].level | 3 |
| concepts[15].score | 0.19842979311943054 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q339011 |
| concepts[15].display_name | Symmetric matrix |
| concepts[16].id | https://openalex.org/C134306372 |
| concepts[16].level | 1 |
| concepts[16].score | 0.08624270558357239 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[16].display_name | Mathematical analysis |
| concepts[17].id | https://openalex.org/C121332964 |
| concepts[17].level | 0 |
| concepts[17].score | 0.0 |
| concepts[17].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[17].display_name | Physics |
| concepts[18].id | https://openalex.org/C159985019 |
| concepts[18].level | 1 |
| concepts[18].score | 0.0 |
| concepts[18].wikidata | https://www.wikidata.org/wiki/Q181790 |
| concepts[18].display_name | Composite material |
| concepts[19].id | https://openalex.org/C158693339 |
| concepts[19].level | 2 |
| concepts[19].score | 0.0 |
| concepts[19].wikidata | https://www.wikidata.org/wiki/Q190524 |
| concepts[19].display_name | Eigenvalues and eigenvectors |
| concepts[20].id | https://openalex.org/C114614502 |
| concepts[20].level | 1 |
| concepts[20].score | 0.0 |
| concepts[20].wikidata | https://www.wikidata.org/wiki/Q76592 |
| concepts[20].display_name | Combinatorics |
| concepts[21].id | https://openalex.org/C192562407 |
| concepts[21].level | 0 |
| concepts[21].score | 0.0 |
| concepts[21].wikidata | https://www.wikidata.org/wiki/Q228736 |
| concepts[21].display_name | Materials science |
| concepts[22].id | https://openalex.org/C62520636 |
| concepts[22].level | 1 |
| concepts[22].score | 0.0 |
| concepts[22].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[22].display_name | Quantum mechanics |
| keywords[0].id | https://openalex.org/keywords/gaussian-elimination |
| keywords[0].score | 0.8746613264083862 |
| keywords[0].display_name | Gaussian elimination |
| keywords[1].id | https://openalex.org/keywords/coefficient-matrix |
| keywords[1].score | 0.7010838985443115 |
| keywords[1].display_name | Coefficient matrix |
| keywords[2].id | https://openalex.org/keywords/gaussian |
| keywords[2].score | 0.6544263958930969 |
| keywords[2].display_name | Gaussian |
| keywords[3].id | https://openalex.org/keywords/rank |
| keywords[3].score | 0.6495974659919739 |
| keywords[3].display_name | Rank (graph theory) |
| keywords[4].id | https://openalex.org/keywords/matrix |
| keywords[4].score | 0.5831514000892639 |
| keywords[4].display_name | Matrix (chemical analysis) |
| keywords[5].id | https://openalex.org/keywords/applied-mathematics |
| keywords[5].score | 0.5599634051322937 |
| keywords[5].display_name | Applied mathematics |
| keywords[6].id | https://openalex.org/keywords/linear-system |
| keywords[6].score | 0.5407045483589172 |
| keywords[6].display_name | Linear system |
| keywords[7].id | https://openalex.org/keywords/computer-science |
| keywords[7].score | 0.5079692006111145 |
| keywords[7].display_name | Computer science |
| keywords[8].id | https://openalex.org/keywords/system-of-linear-equations |
| keywords[8].score | 0.48877617716789246 |
| keywords[8].display_name | System of linear equations |
| keywords[9].id | https://openalex.org/keywords/augmented-matrix |
| keywords[9].score | 0.4852454662322998 |
| keywords[9].display_name | Augmented matrix |
| keywords[10].id | https://openalex.org/keywords/singularity |
| keywords[10].score | 0.45986106991767883 |
| keywords[10].display_name | Singularity |
| keywords[11].id | https://openalex.org/keywords/mathematical-optimization |
| keywords[11].score | 0.435200959444046 |
| keywords[11].display_name | Mathematical optimization |
| keywords[12].id | https://openalex.org/keywords/mathematics |
| keywords[12].score | 0.41063418984413147 |
| keywords[12].display_name | Mathematics |
| keywords[13].id | https://openalex.org/keywords/algorithm |
| keywords[13].score | 0.3379117548465729 |
| keywords[13].display_name | Algorithm |
| keywords[14].id | https://openalex.org/keywords/state-transition-matrix |
| keywords[14].score | 0.32817381620407104 |
| keywords[14].display_name | State-transition matrix |
| keywords[15].id | https://openalex.org/keywords/symmetric-matrix |
| keywords[15].score | 0.19842979311943054 |
| keywords[15].display_name | Symmetric matrix |
| keywords[16].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[16].score | 0.08624270558357239 |
| keywords[16].display_name | Mathematical analysis |
| language | en |
| locations[0].id | doi:10.1088/1742-6596/2012/1/012061 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4210187594 |
| locations[0].source.issn | 1742-6588, 1742-6596 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | 1742-6588 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Journal of Physics Conference Series |
| locations[0].source.host_organization | https://openalex.org/P4310320083 |
| locations[0].source.host_organization_name | IOP Publishing |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310320083, https://openalex.org/P4310311669 |
| locations[0].source.host_organization_lineage_names | IOP Publishing, Institute of Physics |
| locations[0].license | cc-by |
| locations[0].pdf_url | https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdf |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Journal of Physics: Conference Series |
| locations[0].landing_page_url | https://doi.org/10.1088/1742-6596/2012/1/012061 |
| indexed_in | crossref |
| authorships[0].author.id | https://openalex.org/A5100641952 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-3201-4963 |
| authorships[0].author.display_name | Haoyu Luo |
| authorships[0].countries | US |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I4210100835 |
| authorships[0].affiliations[0].raw_affiliation_string | Cheshire Academy, Cheshire, Connecticut, 06410, China |
| authorships[0].institutions[0].id | https://openalex.org/I4210100835 |
| authorships[0].institutions[0].ror | https://ror.org/0145t7b41 |
| authorships[0].institutions[0].type | healthcare |
| authorships[0].institutions[0].lineage | https://openalex.org/I4210100835 |
| authorships[0].institutions[0].country_code | US |
| authorships[0].institutions[0].display_name | Cheshire Medical Center |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Haoyu Luo |
| authorships[0].is_corresponding | False |
| authorships[0].raw_affiliation_strings | Cheshire Academy, Cheshire, Connecticut, 06410, China |
| authorships[1].author.id | https://openalex.org/A5071014171 |
| authorships[1].author.orcid | https://orcid.org/0000-0002-9546-7644 |
| authorships[1].author.display_name | Simeng Wu |
| authorships[1].affiliations[0].raw_affiliation_string | School of Dulwich International High School Suzhou, Suzhou, 215000, China |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Simeng Wu |
| authorships[1].is_corresponding | False |
| authorships[1].raw_affiliation_strings | School of Dulwich International High School Suzhou, Suzhou, 215000, China |
| authorships[2].author.id | https://openalex.org/A5048463213 |
| authorships[2].author.orcid | |
| authorships[2].author.display_name | Ningyun Xie |
| authorships[2].countries | CN |
| authorships[2].affiliations[0].institution_ids | https://openalex.org/I4210120696 |
| authorships[2].affiliations[0].raw_affiliation_string | School of Guangdong Country Garden School, Foshan, 528500, China |
| authorships[2].institutions[0].id | https://openalex.org/I4210120696 |
| authorships[2].institutions[0].ror | https://ror.org/01xqdxh54 |
| authorships[2].institutions[0].type | facility |
| authorships[2].institutions[0].lineage | https://openalex.org/I19820366, https://openalex.org/I4210120696 |
| authorships[2].institutions[0].country_code | CN |
| authorships[2].institutions[0].display_name | South China Botanical Garden |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Ningyun Xie |
| authorships[2].is_corresponding | False |
| authorships[2].raw_affiliation_strings | School of Guangdong Country Garden School, Foshan, 528500, China |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdf |
| open_access.oa_status | diamond |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Three methods for solving systems of linear equations: Comparing the advantages and disadvantages |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T10792 |
| primary_topic.field.id | https://openalex.org/fields/17 |
| primary_topic.field.display_name | Computer Science |
| primary_topic.score | 0.8367999792098999 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/1703 |
| primary_topic.subfield.display_name | Computational Theory and Mathematics |
| primary_topic.display_name | Matrix Theory and Algorithms |
| related_works | https://openalex.org/W2491524947, https://openalex.org/W2585717312, https://openalex.org/W2157943528, https://openalex.org/W2359478957, https://openalex.org/W2365527693, https://openalex.org/W3021239762, https://openalex.org/W2506892714, https://openalex.org/W2088075613, https://openalex.org/W2355478383, https://openalex.org/W2015081464 |
| cited_by_count | 2 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 1 |
| counts_by_year[1].year | 2024 |
| counts_by_year[1].cited_by_count | 1 |
| locations_count | 1 |
| best_oa_location.id | doi:10.1088/1742-6596/2012/1/012061 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4210187594 |
| best_oa_location.source.issn | 1742-6588, 1742-6596 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | 1742-6588 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Journal of Physics Conference Series |
| best_oa_location.source.host_organization | https://openalex.org/P4310320083 |
| best_oa_location.source.host_organization_name | IOP Publishing |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310320083, https://openalex.org/P4310311669 |
| best_oa_location.source.host_organization_lineage_names | IOP Publishing, Institute of Physics |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdf |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Journal of Physics: Conference Series |
| best_oa_location.landing_page_url | https://doi.org/10.1088/1742-6596/2012/1/012061 |
| primary_location.id | doi:10.1088/1742-6596/2012/1/012061 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4210187594 |
| primary_location.source.issn | 1742-6588, 1742-6596 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | 1742-6588 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Journal of Physics Conference Series |
| primary_location.source.host_organization | https://openalex.org/P4310320083 |
| primary_location.source.host_organization_name | IOP Publishing |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310320083, https://openalex.org/P4310311669 |
| primary_location.source.host_organization_lineage_names | IOP Publishing, Institute of Physics |
| primary_location.license | cc-by |
| primary_location.pdf_url | https://iopscience.iop.org/article/10.1088/1742-6596/2012/1/012061/pdf |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Journal of Physics: Conference Series |
| primary_location.landing_page_url | https://doi.org/10.1088/1742-6596/2012/1/012061 |
| publication_date | 2021-09-01 |
| publication_year | 2021 |
| referenced_works | https://openalex.org/W2032494394 |
| referenced_works_count | 1 |
| abstract_inverted_index.a | 49, 64 |
| abstract_inverted_index.At | 100 |
| abstract_inverted_index.To | 52 |
| abstract_inverted_index.We | 36 |
| abstract_inverted_index.as | 140 |
| abstract_inverted_index.be | 34, 93 |
| abstract_inverted_index.in | 27, 132 |
| abstract_inverted_index.no | 82 |
| abstract_inverted_index.of | 97, 121, 134 |
| abstract_inverted_index.on | 43, 68, 129 |
| abstract_inverted_index.to | 4, 13, 24, 63, 92 |
| abstract_inverted_index.we | 18, 55 |
| abstract_inverted_index.For | 77 |
| abstract_inverted_index.The | 123 |
| abstract_inverted_index.and | 111, 136 |
| abstract_inverted_index.for | 85, 117 |
| abstract_inverted_index.has | 81 |
| abstract_inverted_index.how | 23 |
| abstract_inverted_index.its | 86 |
| abstract_inverted_index.not | 33 |
| abstract_inverted_index.row | 109 |
| abstract_inverted_index.the | 20, 29, 44, 53, 61, 95, 101, 115, 119 |
| abstract_inverted_index.way | 116 |
| abstract_inverted_index.This | 1 |
| abstract_inverted_index.aims | 3 |
| abstract_inverted_index.been | 11 |
| abstract_inverted_index.end, | 54 |
| abstract_inverted_index.from | 48 |
| abstract_inverted_index.have | 10, 73 |
| abstract_inverted_index.rank | 120 |
| abstract_inverted_index.rule | 90 |
| abstract_inverted_index.same | 102 |
| abstract_inverted_index.some | 6 |
| abstract_inverted_index.such | 139 |
| abstract_inverted_index.then | 37 |
| abstract_inverted_index.used | 12 |
| abstract_inverted_index.work | 42 |
| abstract_inverted_index.These | 71 |
| abstract_inverted_index.based | 67 |
| abstract_inverted_index.needs | 91 |
| abstract_inverted_index.paper | 2 |
| abstract_inverted_index.paves | 114 |
| abstract_inverted_index.rule, | 58 |
| abstract_inverted_index.solve | 14 |
| abstract_inverted_index.terms | 133 |
| abstract_inverted_index.their | 74 |
| abstract_inverted_index.these | 126 |
| abstract_inverted_index.three | 107 |
| abstract_inverted_index.time, | 103 |
| abstract_inverted_index.under | 94 |
| abstract_inverted_index.which | 9, 28, 41, 59 |
| abstract_inverted_index.would | 32 |
| abstract_inverted_index.choice | 124 |
| abstract_inverted_index.linear | 15, 50, 65 |
| abstract_inverted_index.matrix | 46, 69, 137 |
| abstract_inverted_index.square | 98 |
| abstract_inverted_index.system | 66 |
| abstract_inverted_index.between | 125 |
| abstract_inverted_index.depends | 128 |
| abstract_inverted_index.derived | 47 |
| abstract_inverted_index.limited | 83 |
| abstract_inverted_index.method: | 22 |
| abstract_inverted_index.methods | 72, 127 |
| abstract_inverted_index.present | 56 |
| abstract_inverted_index.system. | 51 |
| abstract_inverted_index.systems | 31 |
| abstract_inverted_index.Abstract | 0 |
| abstract_inverted_index.Firstly, | 17 |
| abstract_inverted_index.Gaussian | 39, 104, 112 |
| abstract_inverted_index.However, | 88 |
| abstract_inverted_index.changed. | 35 |
| abstract_inverted_index.computes | 60 |
| abstract_inverted_index.matrices | 131 |
| abstract_inverted_index.original | 30 |
| abstract_inverted_index.requires | 106 |
| abstract_inverted_index.simplest | 21 |
| abstract_inverted_index.solution | 62 |
| abstract_inverted_index.systems. | 16 |
| abstract_inverted_index.augmented | 45 |
| abstract_inverted_index.computing | 118 |
| abstract_inverted_index.condition | 84, 96 |
| abstract_inverted_index.eliminate | 25 |
| abstract_inverted_index.instance, | 78 |
| abstract_inverted_index.introduce | 5, 19, 38 |
| abstract_inverted_index.matrices. | 99, 122 |
| abstract_inverted_index.prevalent | 7 |
| abstract_inverted_index.variables | 26, 80 |
| abstract_inverted_index.Cramer’s | 57, 89 |
| abstract_inverted_index.dimensions | 135 |
| abstract_inverted_index.elementary | 108 |
| abstract_inverted_index.properties | 138 |
| abstract_inverted_index.scenarios. | 76 |
| abstract_inverted_index.techniques | 8 |
| abstract_inverted_index.application | 75 |
| abstract_inverted_index.coefficient | 130 |
| abstract_inverted_index.eliminating | 79 |
| abstract_inverted_index.elimination | 105, 113 |
| abstract_inverted_index.operations, | 110 |
| abstract_inverted_index.operations. | 87 |
| abstract_inverted_index.singularity. | 141 |
| abstract_inverted_index.determinants. | 70 |
| abstract_inverted_index.eliminations, | 40 |
| cited_by_percentile_year.max | 95 |
| cited_by_percentile_year.min | 90 |
| countries_distinct_count | 2 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile.value | 0.53383985 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |