808:
518:
4087:
803:{\displaystyle {\begin{bmatrix}X&X&X&\cdot &\cdot &\cdot &\cdot &\\X&X&\cdot &X&X&\cdot &\cdot &\\X&\cdot &X&\cdot &X&\cdot &\cdot &\\\cdot &X&\cdot &X&\cdot &X&\cdot &\\\cdot &X&X&\cdot &X&X&X&\\\cdot &\cdot &\cdot &X&X&X&\cdot &\\\cdot &\cdot &\cdot &\cdot &X&\cdot &X&\\\end{bmatrix}}}
277:
259:
1062:
48:
1794:
912:
1271:
In the case of a sparse matrix, substantial memory requirement reductions can be realized by storing only the non-zero entries. Depending on the number and distribution of the non-zero entries, different data structures can be used and yield huge savings in memory when compared to the basic approach.
2216:
The WSE is the largest chip ever made at 46,225 square millimeters in area, it is 56.7 times larger than the largest graphics processing unit. It contains 78 times more AI optimized compute cores, 3,000 times more high speed, on-chip memory, 10,000 times more memory bandwidth, and 33,000 times more
1929:
indexes where each column starts. The name is based on the fact that column index information is compressed relative to the COO format. One typically uses another format (LIL, DOK, COO) for construction. This format is efficient for arithmetic operations, column slicing, and matrix-vector products.
1303:
to the value of the elements. Elements that are missing from the dictionary are taken to be zero. The format is good for incrementally constructing a sparse matrix in random order, but poor for iterating over non-zero values in lexicographical order. One typically constructs a matrix in this format
339:
Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system as only adjacent balls are coupled. By contrast, if the same line of balls were to have springs connecting each ball
379:
that take advantage of the sparse structure of the matrix. Specialized computers have been made for sparse matrices, as they are common in the machine learning field. Operations using standard dense-matrix structures and algorithms are slow and inefficient when applied to large sparse matrices as
2158:
The computation kernel of DNN is large sparse-dense matrix multiplication. In the field of numerical analysis, a sparse matrix is a matrix populated primarily with zeros as elements of the table. By contrast, if the number of non-zero elements in a matrix is relatively large, then it is commonly
1120:
in use. Orthogonalization methods (such as QR factorization) are common, for example, when solving problems by least squares methods. While the theoretical fill-in is still the same, in practical terms the "false non-zeros" can be different for different methods. And symbolic versions of those
1104:
of a matrix are those entries that change from an initial zero to a non-zero value during the execution of an algorithm. To reduce the memory requirements and the number of arithmetic operations used during an algorithm, it is useful to minimize the fill-in by switching rows and columns in the
812:
Matrices with reasonably small upper and lower bandwidth are known as band matrices and often lend themselves to simpler algorithms than general sparse matrices; or one can sometimes apply dense matrix algorithms and gain efficiency simply by looping over a reduced number of indices.
2159:
considered a dense matrix. The fraction of zero elements (non-zero elements) in a matrix is called the sparsity (density). Operations using standard dense-matrix structures and algorithms are relatively slow and consume large amounts of memory when applied to large sparse matrices.
1581:
1642:
254:{\displaystyle \left({\begin{smallmatrix}11&22&0&0&0&0&0\\0&33&44&0&0&0&0\\0&0&55&66&77&0&0\\0&0&0&0&0&88&0\\0&0&0&0&0&0&99\\\end{smallmatrix}}\right)}
1347:
by three (one-dimensional) arrays, that respectively contain nonzero values, the extents of rows, and column indices. It is similar to COO, but compresses the row indices, hence the name. This format allows fast row access and matrix-vector multiplications
2486:(This book, by a professor at the State University of New York at Stony Book, was the first book exclusively dedicated to Sparse Matrices. Graduate courses using this as a textbook were offered at that University in the early 1980s).
1312:
LIL stores one list per row, with each entry containing the column index and the value. Typically, these entries are kept sorted by column index for faster lookup. This is another format good for incremental matrix construction.
1057:{\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{bmatrix}},}
2187:
The WSE contains 400,000 AI-optimized compute cores. Called SLAC™ for Sparse Linear
Algebra Cores, the compute cores are flexible, programmable, and optimized for the sparse linear algebra that underpins all neural network
1469:
1325:
tuples. Ideally, the entries are sorted first by row index and then by column index, to improve random access times. This is another format that is good for incremental matrix construction.
312:
but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered
1789:{\displaystyle {\begin{pmatrix}10&20&0&0&0&0\\0&30&0&40&0&0\\0&0&50&60&70&0\\0&0&0&0&0&80\\\end{pmatrix}}}
1946:
1880:
The (old and new) Yale sparse matrix formats are instances of the CSR scheme. The old Yale format works exactly as described above, with three arrays; the new format combines
1279:
Those that support efficient modification, such as DOK (Dictionary of keys), LIL (List of lists), or COO (Coordinate list). These are typically used to construct the matrices.
4135:
1272:
The trade-off is that accessing the individual elements becomes more complex and additional structures are needed to be able to recover the original matrix unambiguously.
3745:
1172:
1901:
It is likely known as the Yale format because it was proposed in the 1977 Yale Sparse Matrix
Package report from Department of Computer Science at Yale University.
2200:
1867:
would not be redundant). Nonetheless, this does avoid the need to handle an exceptional case when computing the length of each row, as it guarantees the formula
2171:
1192:
3132:
1909:
CSC is similar to CSR except that values are read first by column, a row index is stored for each value, and column pointers are stored. For example, CSC is
4256:
515:. As another example, the following sparse matrix has lower and upper bandwidth both equal to 3. Notice that zeros are represented with dots for clarity.
3959:
4281:
2808:
3178:
4050:
308:
in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as
4276:
4128:
3102:
2768:
2733:
2515:
2450:
2423:
2151:
1955:, a large C++ library, with sub-libraries dedicated to the storage of dense and sparse matrices and solution of corresponding linear systems.
3969:
3735:
1624:
In this case the CSR representation contains 13 entries, compared to 16 in the original matrix. The CSR format saves on memory only when
2201:"Argonne National Laboratory Deploys Cerebras CS-1, the World's Fastest Artificial Intelligence Computer | Argonne National Laboratory"
4235:
4121:
3041:
2471:
2365:
1898:, the data array can be omitted, as the existence of an entry in the row array is sufficient to model a binary adjacency relation.
1576:{\displaystyle {\begin{pmatrix}5&0&0&0\\0&8&0&0\\0&0&3&0\\0&6&0&0\\\end{pmatrix}}}
4266:
1942:
Many software libraries support sparse matrices, and provide solvers for sparse matrix equations. The following are open-source:
1282:
Those that support efficient access and matrix operations, such as CSR (Compressed Sparse Row) or CSC (Compressed Sparse Column).
4193:
3770:
2831:
1863:, so they are in some sense redundant (although in programming languages where the array length needs to be explicitly stored,
1106:
268:
The above sparse matrix contains only 9 non-zero elements, with 26 zero elements. Its sparsity is 74%, and its density is 26%.
3317:
2836:
2110:
2801:
361:
4225:
3534:
3171:
2134:
Yan, Di; Wu, Tao; Liu, Ying; Gao, Yang (2017). "An efficient sparse-dense matrix multiplication on a multicore system".
1357:). The CSR format has been in use since at least the mid-1960s, with the first complete description appearing in 1967.
3609:
2915:
2898:
2000:
416:
4179:
3765:
3287:
3114:
2881:
2876:
2257:
2243:
2229:
2105:
2006:
4230:
3869:
3740:
3654:
2871:
2535:
4302:
4144:
3974:
3864:
3572:
3252:
2910:
2905:
2864:
2794:
1877:. Moreover, the memory cost of this redundant storage is likely insignificant for a sufficiently large matrix.
352:, which typically have a low density of significant data or connections. Large sparse matrices often appear in
4009:
3938:
3820:
3680:
3277:
3164:
3145:
3122:
31:
3879:
3462:
3267:
3127:
2927:
2660:
2285:
2003:, another finite element library that also has a sub-library for sparse linear systems and their solution.
1117:
1110:
4189:
3825:
3562:
3412:
3407:
3242:
3217:
3212:
3053:
3008:
2970:
2070:
1958:
900:
852:
389:
340:
to all other balls, the system would correspond to a dense matrix. The concept of sparsity is useful in
281:
4086:
2278:
Parallel sparse matrix-vector and matrix-transpose-vector multiplication using compressed sparse blocks
1206:
A matrix is typically stored as a two-dimensional array. Each entry in the array represents an element
392:. Some very large sparse matrices are infeasible to manipulate using standard dense-matrix algorithms.
1949:, a large C library, containing many different matrix solvers for a variety of matrix storage formats.
4184:
4019:
3377:
3207:
3187:
2993:
2544:
2272:
1391:
1222:
305:
293:
2665:
2290:
1997:, a finite element library that also has a sub-library for sparse linear systems and their solution.
4163:
4040:
4014:
3592:
3397:
3387:
2411:
2085:
3036:
2027:
Fortran 77 library for sparse matrix diagonalization and manipulation, using the
Arnoldi algorithm
1121:
algorithms can be used in the same manner as the symbolic
Cholesky to compute worst case fill-in.
4091:
4045:
4035:
3989:
3984:
3913:
3849:
3715:
3452:
3447:
3382:
3372:
3237:
3021:
2886:
2846:
2635:
2570:
2340:
1962:
1137:
504:
349:
289:
2579:
Also NOAA Technical
Memorandum NOS NGS-4, National Geodetic Survey, Rockville, MD. Referencing
276:
4102:
3889:
3884:
3874:
3854:
3815:
3810:
3639:
3634:
3619:
3614:
3605:
3600:
3547:
3442:
3392:
3337:
3307:
3302:
3282:
3272:
3232:
2955:
2854:
2774:
2764:
2739:
2729:
2695:
A EU-funded project on sparse models, algorithms and dictionary learning for large-scale data.
2521:
2511:
2477:
2467:
2446:
2419:
2147:
2095:
1292:
4199:
4097:
4065:
3994:
3933:
3928:
3908:
3844:
3750:
3720:
3705:
3690:
3685:
3624:
3577:
3552:
3542:
3513:
3432:
3427:
3402:
3332:
3312:
3222:
3202:
2978:
2756:
2721:
2704:
2670:
2627:
2592:
2560:
2552:
2438:
2332:
2139:
2075:
2040:
1968:
1895:
1258:
matrix, the amount of memory required to store the matrix in this format is proportional to
1147:
1130:
884:
880:
385:
2461:
4307:
4240:
3795:
3730:
3710:
3695:
3675:
3659:
3557:
3488:
3478:
3437:
3322:
3292:
2998:
2940:
2057:
843:
830:
381:
2548:
2358:
1961:
is a C++ library that contains several sparse matrix solvers. However, none of them are
4158:
4055:
3999:
3979:
3964:
3923:
3800:
3760:
3725:
3649:
3588:
3567:
3508:
3498:
3483:
3417:
3362:
3352:
3347:
3257:
3090:
3068:
2893:
2817:
2504:
2090:
1988:
1892:
1195:
1177:
888:
376:
345:
2491:
2317:
2276:
1917:
is an array of the (top-to-bottom, then left-to-right) non-zero values of the matrix;
1602:
Then we take slices from V and COL_INDEX starting at row_start and ending at row_end.
4296:
4060:
3918:
3859:
3790:
3780:
3775:
3700:
3629:
3503:
3493:
3342:
3327:
3262:
3063:
2945:
2574:
2407:
2080:
1410:, and contain the non-zero values and the column indices of those values respectively
848:
341:
2639:
316:. The number of zero-valued elements divided by the total number of elements (e.g.,
3943:
3900:
3805:
3518:
3457:
3367:
3247:
2648:
2565:
2344:
2036:
1300:
2681:
1268:(disregarding the fact that the dimensions of the matrix also need to be stored).
1930:
This is the traditional format for specifying a sparse matrix in MATLAB (via the
3785:
3755:
3523:
3357:
3227:
3058:
2983:
2100:
2015:
provides support for several sparse matrix formats, linear algebra, and solvers.
460:
412:
406:
357:
58:
2615:
3836:
3297:
3046:
2950:
2708:
2596:
2533:
Snay, Richard A. (1976). "Reducing the profile of sparse symmetric matrices".
2442:
2357:
Eisenstat, S. C.; Gursky, M. C.; Schultz, M. H.; Sherman, A. H. (April 1977).
2143:
353:
2778:
2743:
2525:
2481:
4220:
4070:
3644:
2988:
2935:
2760:
2725:
1109:
can be used to calculate the worst possible fill-in before doing the actual
903:
consists of sub-matrices along its diagonal blocks. A block-diagonal matrix
372:
4113:
3085:
2631:
2136:
2017 IEEE 17th
International Conference on Communication Technology (ICCT)
1849:(10, 20, ...) (0, 30, 0, 40, ...)(0, 0, 50, 60, 70, 0) (0, 0, 0, 0, 0, 80)
1621:. We now know that in row 1 we have one element at column 1 with value 8.
4004:
3031:
2859:
1952:
368:
2466:. Mathematics in science and engineering. Vol. 99. Academic Press.
2172:"Cerebras Systems Unveils the Industry's First Trillion Transistor Chip"
17:
3080:
3026:
2556:
2336:
1994:
1888:
into a single array and handles the diagonal of the matrix separately.
2033:
Library for solution of large scale linear systems and sparse matrices
4271:
4261:
3075:
3016:
2024:
2018:
841:
A very efficient structure for an extreme case of band matrices, the
2674:
2616:"A comparison of several bandwidth and profile reduction algorithms"
1198:
can significantly accelerate convergence of such iterative methods.
36:
2271:
Buluç, Aydın; Fineman, Jeremy T.; Frigo, Matteo; Gilbert, John R.;
1304:
and then converts to another more efficient format for processing.
2786:
2030:
2012:
1141:
275:
3156:
2614:
Gibbs, Norman E.; Poole, William G.; Stockmeyer, Paul K. (1976).
3097:
829:
with a lower bandwidth. A number of algorithms are designed for
4117:
3160:
2790:
2692:
384:
are wasted on the zeros. Sparse data is by nature more easily
2389:
2284:. ACM Symp. on Parallelism in Algorithms and Architectures.
2060:
who initiated some pioneering work but then left the field.
1605:
To extract the row 1 (the second row) of this matrix we set
284:
in two dimensions. The non-zero elements are shown in black.
2687:
2647:
Gilbert, John R.; Moler, Cleve; Schreiber, Robert (1992).
2021:
is a C++ and C# library with sparse linear algebra support
371:, it is beneficial and often necessary to use specialized
2009:
provides a user-friendly C++ wrapper for BLAS and LAPACK.
1248:
is the column index, numbered from left to right. For an
2129:
2127:
2701:
Iterative
Solution of Large Sparse Systems of Equations
2649:"Sparse matrices in MATLAB: Design and Implementation"
1651:
1478:
929:
527:
1921:
is the row indices corresponding to the values; and,
1645:
1472:
1416:
contains the column in which the corresponding entry
1180:
1150:
915:
521:
51:
1144:
utilize fast computations of matrix-vector products
1133:
and direct methods exist for sparse matrix solving.
4249:
4213:
4172:
4151:
4028:
3952:
3898:
3834:
3668:
3586:
3532:
3471:
3195:
3113:
3007:
2969:
2926:
2845:
2824:
2043:, provides support for sparse matrices and solvers.
1242:is the row index, numbered from top to bottom, and
367:When storing and manipulating sparse matrices on a
2503:
1788:
1575:
1442:where the given row starts. This is equivalent to
1186:
1166:
1056:
802:
253:
1376:in row form using three (one-dimensional) arrays
2653:SIAM Journal on Matrix Analysis and Applications
1800:matrix (24 entries) with 8 nonzero elements, so
1446:encoding the total number of nonzeros above row
816:By rearranging the rows and columns of a matrix
411:An important special type of sparse matrices is
1329:Compressed sparse row (CSR, CRS or Yale format)
30:"Sparsity" redirects here. For other uses, see
2390:Oral history interview with Harry M. Markowitz
4129:
3172:
2802:
2318:"Sparse Matrix Multiplication Package (SMMP)"
2316:Bank, Randolph E.; Douglas, Craig C. (1993),
1855:Note that in this format, the first value of
500:
27:Matrix in which most of the elements are zero
8:
1599:row_start = ROW_INDEX row_end = ROW_INDEX
2718:Iterative Methods for Sparse Linear Systems
4136:
4122:
4114:
3746:Fundamental (linear differential equation)
3179:
3165:
3157:
2809:
2795:
2787:
2418:(3rd ed.). Baltimore: Johns Hopkins.
1803:V = COL_INDEX = ROW_INDEX =
2664:
2620:ACM Transactions on Mathematical Software
2564:
2289:
1646:
1644:
1473:
1471:
1341:(CRS) or Yale format represents a matrix
1221:of the matrix and is accessed by the two
1179:
1158:
1149:
1037:
1032:
974:
969:
938:
933:
924:
916:
914:
522:
520:
56:
50:
2753:Direct Methods for Sparse Linear Systems
2587:Scott, Jennifer; Tuma, Miroslav (2023).
1806:The whole is stored as 21 entries: 8 in
1590:V = COL_INDEX = ROW_INDEX =
1384:denote the number of nonzero entries in
1275:Formats can be divided into two groups:
879:A symmetric sparse matrix arises as the
332:matrix) is sometimes referred to as the
280:A sparse matrix obtained when solving a
4267:Basic Linear Algebra Subprograms (BLAS)
4051:Matrix representation of conic sections
2433:Stoer, Josef; Bulirsch, Roland (2002).
2123:
1458:immediately after the last valid index
2492:"Sparse Matrix Multiplication Package"
1859:is always zero and the last is always
1587:matrix with 4 nonzero elements, hence
847:, is to store just the entries in the
822:it may be possible to obtain a matrix
2490:Bank, Randolph E.; Douglas, Craig C.
2325:Advances in Computational Mathematics
1905:Compressed sparse column (CSC or CCS)
887:; it can be stored efficiently as an
388:and thus requires significantly less
7:
2589:Algorithms for Sparse Linear Systems
2580:
2304:
1987:olver), written in Fortran90, is a
1831:(10, 20) (30, 40) (50, 60, 70) (80)
1596:To extract a row, we first define:
2435:Introduction to Numerical Analysis
2371:from the original on April 6, 2019
1593:assuming a zero-indexed language.
25:
2682:Sparse Matrix Algorithms Research
1841:) where each row starts and ends;
1116:There are other methods than the
57:
4085:
2684:at the Texas A&M University.
1454:, i.e., the fictitious index in
1033:
970:
934:
917:
3953:Used in science and engineering
1360:The CSR format stores a sparse
1125:Solving sparse matrix equations
1107:symbolic Cholesky decomposition
3196:Explicitly constrained entries
2460:Tewarson, Reginald P. (1973).
2111:Matrix Market exchange formats
362:partial differential equations
344:and application areas such as
1:
3970:Fundamental (computer vision)
2688:SuiteSparse Matrix Collection
2699:Hackbusch, Wolfgang (2016).
2502:Pissanetzky, Sergio (1984).
2359:"Yale Sparse Matrix Package"
1639:Another example, the matrix
3736:Duplication and elimination
3535:eigenvalues or eigenvectors
3133:Directed acyclic word graph
2899:Double-ended priority queue
1136:Iterative methods, such as
1077:is a square matrix for all
417:lower bandwidth of a matrix
4324:
4180:System of linear equations
3669:With specific applications
3298:Discrete Fourier Transform
2751:Davis, Timothy A. (2006).
2703:(2nd ed.). Springer.
2437:(3rd ed.). Springer.
2106:Harwell-Boeing file format
1847:aligns values in columns:
1833:, indicating the index of
1613:. Then we make the slices
503:, §1.2.1). For example, a
415:, defined as follows. The
404:
360:applications when solving
29:
4231:Cache-oblivious algorithm
4079:
3960:Cabibbo–Kobayashi–Maskawa
3587:Satisfying conditions on
3141:
2709:10.1007/978-3-319-28483-5
2597:10.1007/978-3-031-25820-6
2443:10.1007/978-0-387-21738-3
2144:10.1109/icct.2017.8359956
2138:. IEEE. pp. 1880–3.
1434:and encodes the index in
1378:(V, COL_INDEX, ROW_INDEX)
501:Golub & Van Loan 1996
4282:General purpose software
4145:Numerical linear algebra
2865:Retrieval Data Structure
2506:Sparse Matrix Technology
2217:communication bandwidth.
1466:For example, the matrix
1287:Dictionary of keys (DOK)
42:Example of sparse matrix
3318:Generalized permutation
3146:List of data structures
3123:Binary decision diagram
2761:10.1137/1.9780898718881
2726:10.1137/1.9780898718003
2566:2027/uc1.31210024848523
2259:scipy.sparse.coo_matrix
2245:scipy.sparse.lil_matrix
2231:scipy.sparse.dok_matrix
2056:was possibly coined by
2039:, a Python library for
1911:(val, row_ind, col_ptr)
1450:. The last element is
463:is the smallest number
424:is the smallest number
32:Sparse (disambiguation)
4092:Mathematics portal
3128:Directed acyclic graph
1983:arallel sparse direct
1790:
1577:
1339:compressed row storage
1194:is sparse. The use of
1188:
1168:
1167:{\displaystyle Ax_{i}}
1118:Cholesky decomposition
1111:Cholesky decomposition
1058:
831:bandwidth minimization
804:
285:
282:finite element problem
255:
4277:Specialized libraries
4190:Matrix multiplication
4185:Matrix decompositions
2716:Saad, Yousef (2003).
2632:10.1145/355705.355707
2273:Leiserson, Charles E.
2071:Matrix representation
1869:ROW_INDEX − ROW_INDEX
1791:
1578:
1394:shall be used here.)
1335:compressed sparse row
1321:COO stores a list of
1317:Coordinate list (COO)
1189:
1169:
1059:
901:block-diagonal matrix
865:matrix requires only
853:one-dimensional array
805:
279:
256:
2994:Unrolled linked list
2412:Van Loan, Charles F.
2176:www.businesswire.com
1643:
1470:
1323:(row, column, value)
1178:
1148:
913:
519:
511:and upper bandwidth
507:has lower bandwidth
430:such that the entry
294:scientific computing
49:
4164:Numerical stability
4041:Linear independence
3288:Diagonally dominant
3042:Self-balancing tree
2549:1976BGeod..50..341S
2536:Bulletin Géodésique
2416:Matrix Computations
2086:Single-entry matrix
1308:List of lists (LIL)
4046:Matrix exponential
4036:Jordan normal form
3870:Fisher information
3741:Euclidean distance
3655:Totally unimodular
3022:Binary search tree
2887:Double-ended queue
2557:10.1007/BF02521587
2510:. Academic Press.
2337:10.1007/BF02070824
1896:adjacency matrices
1871:works for any row
1786:
1780:
1573:
1567:
1392:zero-based indices
1291:DOK consists of a
1236:. Conventionally,
1184:
1164:
1138:conjugate gradient
1054:
1045:
800:
794:
505:tridiagonal matrix
445:vanishes whenever
350:numerical analysis
290:numerical analysis
286:
251:
245:
244:
4290:
4289:
4111:
4110:
4103:Category:Matrices
3975:Fuzzy associative
3865:Doubly stochastic
3573:Positive-definite
3253:Block tridiagonal
3154:
3153:
2956:Hashed array tree
2855:Associative array
2770:978-0-89871-613-9
2735:978-0-89871-534-7
2517:978-0-12-557580-5
2452:978-0-387-95452-3
2425:978-0-8018-5414-9
2392:, pp. 9, 10.
2153:978-1-5090-3944-9
2096:Sparse graph code
1825:splits the array
1187:{\displaystyle A}
459:. Similarly, the
274:
273:
16:(Redirected from
4315:
4200:Matrix splitting
4138:
4131:
4124:
4115:
4098:List of matrices
4090:
4089:
4066:Row echelon form
4010:State transition
3939:Seidel adjacency
3821:Totally positive
3681:Alternating sign
3278:Complex Hadamard
3181:
3174:
3167:
3158:
2979:Association list
2811:
2804:
2797:
2788:
2782:
2747:
2712:
2678:
2668:
2643:
2600:
2578:
2568:
2529:
2509:
2498:
2496:
2485:
2456:
2429:
2393:
2387:
2381:
2380:
2378:
2376:
2370:
2363:
2354:
2348:
2347:
2322:
2313:
2307:
2302:
2296:
2295:
2293:
2283:
2268:
2262:
2260:
2254:
2248:
2246:
2240:
2234:
2232:
2226:
2220:
2219:
2213:
2212:
2197:
2191:
2190:
2184:
2183:
2168:
2162:
2161:
2131:
2076:Pareto principle
2041:machine learning
1933:
1928:
1924:
1920:
1916:
1912:
1887:
1883:
1876:
1870:
1866:
1862:
1858:
1850:
1846:
1840:
1836:
1832:
1828:
1824:
1817:
1813:
1809:
1799:
1795:
1793:
1792:
1787:
1785:
1784:
1635:
1620:
1616:
1612:
1608:
1586:
1582:
1580:
1579:
1574:
1572:
1571:
1461:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1426:
1419:
1415:
1409:
1405:
1401:
1389:
1383:
1379:
1375:
1369:
1356:
1346:
1324:
1298:
1267:
1257:
1247:
1241:
1235:
1229:
1220:
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1163:
1162:
1096:Reducing fill-in
1086:
1076:
1063:
1061:
1060:
1055:
1050:
1049:
1042:
1041:
1036:
979:
978:
973:
943:
942:
937:
920:
908:
885:undirected graph
881:adjacency matrix
870:
864:
855:, so a diagonal
828:
821:
809:
807:
806:
801:
799:
798:
792:
754:
716:
678:
640:
602:
564:
514:
510:
498:
484:
468:
458:
444:
429:
423:
269:
260:
258:
257:
252:
250:
246:
37:
21:
4323:
4322:
4318:
4317:
4316:
4314:
4313:
4312:
4303:Sparse matrices
4293:
4292:
4291:
4286:
4245:
4241:Multiprocessing
4209:
4205:Sparse problems
4168:
4147:
4142:
4112:
4107:
4084:
4075:
4024:
3948:
3894:
3830:
3664:
3582:
3528:
3467:
3268:Centrosymmetric
3191:
3185:
3155:
3150:
3137:
3109:
3003:
2999:XOR linked list
2965:
2941:Circular buffer
2922:
2841:
2820:
2818:Data structures
2815:
2785:
2771:
2750:
2736:
2715:
2698:
2675:10.1137/0613024
2666:10.1.1.470.1054
2646:
2613:
2609:
2607:Further reading
2604:
2586:
2532:
2518:
2501:
2494:
2489:
2474:
2463:Sparse Matrices
2459:
2453:
2432:
2426:
2406:
2402:
2397:
2396:
2388:
2384:
2374:
2372:
2368:
2361:
2356:
2355:
2351:
2320:
2315:
2314:
2310:
2303:
2299:
2291:10.1.1.211.5256
2281:
2270:
2269:
2265:
2258:
2255:
2251:
2244:
2241:
2237:
2230:
2227:
2223:
2210:
2208:
2207:(Press release)
2199:
2198:
2194:
2181:
2179:
2170:
2169:
2165:
2154:
2133:
2132:
2125:
2120:
2115:
2066:
2058:Harry Markowitz
2050:
1940:
1931:
1926:
1925:is the list of
1922:
1918:
1914:
1910:
1907:
1885:
1881:
1872:
1868:
1864:
1860:
1856:
1848:
1844:
1838:
1834:
1830:
1826:
1822:
1815:
1811:
1807:
1804:
1797:
1779:
1778:
1773:
1768:
1763:
1758:
1753:
1747:
1746:
1741:
1736:
1731:
1726:
1721:
1715:
1714:
1709:
1704:
1699:
1694:
1689:
1683:
1682:
1677:
1672:
1667:
1662:
1657:
1647:
1641:
1640:
1625:
1618:
1614:
1610:
1606:
1600:
1591:
1584:
1566:
1565:
1560:
1555:
1550:
1544:
1543:
1538:
1533:
1528:
1522:
1521:
1516:
1511:
1506:
1500:
1499:
1494:
1489:
1484:
1474:
1468:
1467:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1428:
1424:
1417:
1413:
1407:
1403:
1399:
1385:
1381:
1377:
1371:
1361:
1349:
1342:
1331:
1322:
1319:
1310:
1296:
1289:
1259:
1249:
1243:
1237:
1231:
1225:
1219:
1207:
1204:
1196:preconditioners
1176:
1175:
1174:, where matrix
1154:
1146:
1145:
1127:
1098:
1093:
1078:
1075:
1067:
1044:
1043:
1031:
1029:
1024:
1019:
1013:
1012:
1007:
1002:
997:
991:
990:
985:
980:
968:
966:
960:
959:
954:
949:
944:
932:
925:
911:
910:
904:
897:
877:
866:
856:
844:diagonal matrix
839:
823:
817:
793:
791:
786:
781:
776:
771:
766:
761:
755:
753:
748:
743:
738:
733:
728:
723:
717:
715:
710:
705:
700:
695:
690:
685:
679:
677:
672:
667:
662:
657:
652:
647:
641:
639:
634:
629:
624:
619:
614:
609:
603:
601:
596:
591:
586:
581:
576:
571:
565:
563:
558:
553:
548:
543:
538:
533:
523:
517:
516:
512:
508:
486:
482:
470:
464:
461:upper bandwidth
446:
443:
431:
425:
419:
409:
403:
398:
380:processing and
377:data structures
336:of the matrix.
270:
267:
261:
243:
242:
237:
232:
227:
222:
217:
212:
206:
205:
200:
195:
190:
185:
180:
175:
169:
168:
163:
158:
153:
148:
143:
138:
132:
131:
126:
121:
116:
111:
106:
101:
95:
94:
89:
84:
79:
74:
69:
64:
52:
47:
46:
44:
35:
28:
23:
22:
15:
12:
11:
5:
4321:
4319:
4311:
4310:
4305:
4295:
4294:
4288:
4287:
4285:
4284:
4279:
4274:
4269:
4264:
4259:
4253:
4251:
4247:
4246:
4244:
4243:
4238:
4233:
4228:
4223:
4217:
4215:
4211:
4210:
4208:
4207:
4202:
4197:
4187:
4182:
4176:
4174:
4170:
4169:
4167:
4166:
4161:
4159:Floating point
4155:
4153:
4149:
4148:
4143:
4141:
4140:
4133:
4126:
4118:
4109:
4108:
4106:
4105:
4100:
4095:
4080:
4077:
4076:
4074:
4073:
4068:
4063:
4058:
4056:Perfect matrix
4053:
4048:
4043:
4038:
4032:
4030:
4026:
4025:
4023:
4022:
4017:
4012:
4007:
4002:
3997:
3992:
3987:
3982:
3977:
3972:
3967:
3962:
3956:
3954:
3950:
3949:
3947:
3946:
3941:
3936:
3931:
3926:
3921:
3916:
3911:
3905:
3903:
3896:
3895:
3893:
3892:
3887:
3882:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3841:
3839:
3832:
3831:
3829:
3828:
3826:Transformation
3823:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3773:
3768:
3763:
3758:
3753:
3748:
3743:
3738:
3733:
3728:
3723:
3718:
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3678:
3672:
3670:
3666:
3665:
3663:
3662:
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3603:
3597:
3595:
3584:
3583:
3581:
3580:
3575:
3570:
3565:
3563:Diagonalizable
3560:
3555:
3550:
3545:
3539:
3537:
3533:Conditions on
3530:
3529:
3527:
3526:
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3486:
3481:
3475:
3473:
3469:
3468:
3466:
3465:
3460:
3455:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3415:
3413:Skew-symmetric
3410:
3408:Skew-Hermitian
3405:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3300:
3295:
3290:
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3243:Block-diagonal
3240:
3235:
3230:
3225:
3220:
3218:Anti-symmetric
3215:
3213:Anti-Hermitian
3210:
3205:
3199:
3197:
3193:
3192:
3186:
3184:
3183:
3176:
3169:
3161:
3152:
3151:
3149:
3148:
3142:
3139:
3138:
3136:
3135:
3130:
3125:
3119:
3117:
3111:
3110:
3108:
3107:
3106:
3105:
3095:
3094:
3093:
3091:Hilbert R-tree
3088:
3083:
3073:
3072:
3071:
3069:Fibonacci heap
3066:
3061:
3051:
3050:
3049:
3044:
3039:
3037:Red–black tree
3034:
3029:
3019:
3013:
3011:
3005:
3004:
3002:
3001:
2996:
2991:
2986:
2981:
2975:
2973:
2967:
2966:
2964:
2963:
2958:
2953:
2948:
2943:
2938:
2932:
2930:
2924:
2923:
2921:
2920:
2919:
2918:
2913:
2903:
2902:
2901:
2894:Priority queue
2891:
2890:
2889:
2879:
2874:
2869:
2868:
2867:
2862:
2851:
2849:
2843:
2842:
2840:
2839:
2834:
2828:
2826:
2822:
2821:
2816:
2814:
2813:
2806:
2799:
2791:
2784:
2783:
2769:
2748:
2734:
2713:
2696:
2690:
2685:
2679:
2659:(1): 333–356.
2644:
2626:(4): 322–330.
2610:
2608:
2605:
2603:
2602:
2591:. Birkhauser.
2584:
2543:(4): 341–352.
2530:
2516:
2499:
2487:
2472:
2457:
2451:
2430:
2424:
2408:Golub, Gene H.
2403:
2401:
2398:
2395:
2394:
2382:
2349:
2308:
2297:
2263:
2249:
2235:
2221:
2192:
2163:
2152:
2122:
2121:
2119:
2116:
2114:
2113:
2108:
2103:
2098:
2093:
2091:Skyline matrix
2088:
2083:
2078:
2073:
2067:
2065:
2062:
2049:
2046:
2045:
2044:
2034:
2028:
2022:
2016:
2010:
2004:
1998:
1992:
1989:frontal solver
1966:
1956:
1950:
1939:
1936:
1906:
1903:
1853:
1852:
1842:
1802:
1783:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1748:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1716:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1688:
1685:
1684:
1681:
1678:
1676:
1673:
1671:
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1652:
1650:
1598:
1589:
1570:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1545:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1523:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1501:
1498:
1495:
1493:
1490:
1488:
1485:
1483:
1480:
1479:
1477:
1464:
1463:
1421:
1411:
1406:are of length
1330:
1327:
1318:
1315:
1309:
1306:
1288:
1285:
1284:
1283:
1280:
1211:
1203:
1200:
1183:
1161:
1157:
1153:
1126:
1123:
1097:
1094:
1092:
1089:
1071:
1053:
1048:
1040:
1035:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1014:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
992:
989:
986:
984:
981:
977:
972:
967:
965:
962:
961:
958:
955:
953:
950:
948:
945:
941:
936:
931:
930:
928:
923:
919:
896:
895:Block diagonal
893:
889:adjacency list
876:
873:
838:
835:
797:
790:
787:
785:
782:
780:
777:
775:
772:
770:
767:
765:
762:
760:
757:
756:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
718:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
680:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
642:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
604:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
570:
567:
566:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
528:
526:
474:
435:
405:Main article:
402:
399:
397:
394:
346:network theory
272:
271:
266:
263:
262:
249:
241:
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
207:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
170:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
137:
134:
133:
130:
127:
125:
122:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
96:
93:
90:
88:
85:
83:
80:
78:
75:
73:
70:
68:
65:
63:
60:
59:
55:
45:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4320:
4309:
4306:
4304:
4301:
4300:
4298:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4254:
4252:
4248:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4218:
4216:
4212:
4206:
4203:
4201:
4198:
4195:
4191:
4188:
4186:
4183:
4181:
4178:
4177:
4175:
4171:
4165:
4162:
4160:
4157:
4156:
4154:
4150:
4146:
4139:
4134:
4132:
4127:
4125:
4120:
4119:
4116:
4104:
4101:
4099:
4096:
4094:
4093:
4088:
4082:
4081:
4078:
4072:
4069:
4067:
4064:
4062:
4061:Pseudoinverse
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4033:
4031:
4029:Related terms
4027:
4021:
4020:Z (chemistry)
4018:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3957:
3955:
3951:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3906:
3904:
3902:
3897:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3842:
3840:
3838:
3833:
3827:
3824:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3673:
3671:
3667:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3607:
3604:
3602:
3599:
3598:
3596:
3594:
3590:
3585:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3540:
3538:
3536:
3531:
3525:
3522:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3476:
3474:
3470:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3378:Pentadiagonal
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3208:Anti-diagonal
3206:
3204:
3201:
3200:
3198:
3194:
3189:
3182:
3177:
3175:
3170:
3168:
3163:
3162:
3159:
3147:
3144:
3143:
3140:
3134:
3131:
3129:
3126:
3124:
3121:
3120:
3118:
3116:
3112:
3104:
3101:
3100:
3099:
3096:
3092:
3089:
3087:
3084:
3082:
3079:
3078:
3077:
3074:
3070:
3067:
3065:
3064:Binomial heap
3062:
3060:
3057:
3056:
3055:
3052:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3024:
3023:
3020:
3018:
3015:
3014:
3012:
3010:
3006:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2974:
2972:
2968:
2962:
2961:Sparse matrix
2959:
2957:
2954:
2952:
2949:
2947:
2946:Dynamic array
2944:
2942:
2939:
2937:
2934:
2933:
2931:
2929:
2925:
2917:
2914:
2912:
2909:
2908:
2907:
2904:
2900:
2897:
2896:
2895:
2892:
2888:
2885:
2884:
2883:
2880:
2878:
2875:
2873:
2870:
2866:
2863:
2861:
2858:
2857:
2856:
2853:
2852:
2850:
2848:
2844:
2838:
2835:
2833:
2830:
2829:
2827:
2823:
2819:
2812:
2807:
2805:
2800:
2798:
2793:
2792:
2789:
2780:
2776:
2772:
2766:
2762:
2758:
2754:
2749:
2745:
2741:
2737:
2731:
2727:
2723:
2719:
2714:
2710:
2706:
2702:
2697:
2694:
2693:SMALL project
2691:
2689:
2686:
2683:
2680:
2676:
2672:
2667:
2662:
2658:
2654:
2650:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2612:
2611:
2606:
2601:(Open Access)
2598:
2594:
2590:
2585:
2582:
2576:
2572:
2567:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2537:
2531:
2527:
2523:
2519:
2513:
2508:
2507:
2500:
2493:
2488:
2483:
2479:
2475:
2473:0-12-685650-8
2469:
2465:
2464:
2458:
2454:
2448:
2444:
2440:
2436:
2431:
2427:
2421:
2417:
2413:
2409:
2405:
2404:
2399:
2391:
2386:
2383:
2367:
2360:
2353:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2319:
2312:
2309:
2306:
2301:
2298:
2292:
2287:
2280:
2279:
2274:
2267:
2264:
2261:
2253:
2250:
2247:
2239:
2236:
2233:
2225:
2222:
2218:
2206:
2202:
2196:
2193:
2189:
2177:
2173:
2167:
2164:
2160:
2155:
2149:
2145:
2141:
2137:
2130:
2128:
2124:
2117:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2081:Ragged matrix
2079:
2077:
2074:
2072:
2069:
2068:
2063:
2061:
2059:
2055:
2054:sparse matrix
2047:
2042:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1986:
1982:
1978:
1974:
1970:
1967:
1964:
1960:
1957:
1954:
1951:
1948:
1945:
1944:
1943:
1937:
1935:
1904:
1902:
1899:
1897:
1894:
1889:
1878:
1875:
1843:
1821:
1820:
1819:
1801:
1781:
1775:
1770:
1765:
1760:
1755:
1750:
1743:
1738:
1733:
1728:
1723:
1718:
1711:
1706:
1701:
1696:
1691:
1686:
1679:
1674:
1669:
1664:
1659:
1654:
1648:
1637:
1634:− 1) − 1) / 2
1633:
1629:
1622:
1603:
1597:
1594:
1588:
1568:
1562:
1557:
1552:
1547:
1540:
1535:
1530:
1525:
1518:
1513:
1508:
1503:
1496:
1491:
1486:
1481:
1475:
1431:
1427:is of length
1422:
1412:
1397:
1396:
1395:
1393:
1390:. (Note that
1388:
1374:
1368:
1364:
1358:
1355:
1352:
1345:
1340:
1336:
1328:
1326:
1316:
1314:
1307:
1305:
1302:
1297:(row, column)
1294:
1286:
1281:
1278:
1277:
1276:
1273:
1269:
1266:
1262:
1256:
1252:
1246:
1240:
1234:
1228:
1224:
1218:
1214:
1210:
1201:
1199:
1197:
1181:
1159:
1155:
1151:
1143:
1139:
1134:
1132:
1124:
1122:
1119:
1114:
1112:
1108:
1103:
1095:
1090:
1088:
1085:
1081:
1074:
1070:
1064:
1051:
1046:
1038:
1026:
1021:
1016:
1009:
1004:
999:
994:
987:
982:
975:
963:
956:
951:
946:
939:
926:
921:
909:has the form
907:
902:
894:
892:
890:
886:
882:
874:
872:
869:
863:
859:
854:
850:
849:main diagonal
846:
845:
836:
834:
832:
826:
820:
814:
810:
795:
788:
783:
778:
773:
768:
763:
758:
750:
745:
740:
735:
730:
725:
720:
712:
707:
702:
697:
692:
687:
682:
674:
669:
664:
659:
654:
649:
644:
636:
631:
626:
621:
616:
611:
606:
598:
593:
588:
583:
578:
573:
568:
560:
555:
550:
545:
540:
535:
530:
524:
506:
502:
497:
493:
489:
481:
477:
473:
467:
462:
457:
453:
449:
442:
438:
434:
428:
422:
418:
414:
408:
400:
396:Special cases
395:
393:
391:
387:
383:
378:
374:
370:
365:
363:
359:
355:
351:
347:
343:
342:combinatorics
337:
335:
331:
327:
323:
319:
315:
311:
307:
303:
299:
298:sparse matrix
295:
291:
283:
278:
265:
264:
247:
239:
234:
229:
224:
219:
214:
209:
202:
197:
192:
187:
182:
177:
172:
165:
160:
155:
150:
145:
140:
135:
128:
123:
118:
113:
108:
103:
98:
91:
86:
81:
76:
71:
66:
61:
53:
43:
39:
38:
33:
19:
4204:
4152:Key concepts
4083:
4015:Substitution
3901:graph theory
3422:
3398:Quaternionic
3388:Persymmetric
2960:
2916:Disjoint-set
2752:
2717:
2700:
2656:
2652:
2623:
2619:
2588:
2540:
2534:
2505:
2462:
2434:
2415:
2385:
2373:. Retrieved
2352:
2328:
2324:
2311:
2300:
2277:
2266:
2252:
2238:
2224:
2215:
2209:. Retrieved
2204:
2195:
2186:
2180:. Retrieved
2178:. 2019-08-19
2175:
2166:
2157:
2135:
2053:
2051:
2037:scikit-learn
1984:
1980:
1976:
1972:
1963:parallelized
1941:
1908:
1900:
1890:
1879:
1873:
1854:
1805:
1638:
1631:
1627:
1623:
1619:COL_INDEX =
1604:
1601:
1595:
1592:
1465:
1429:
1386:
1372:
1366:
1362:
1359:
1353:
1350:
1343:
1338:
1334:
1332:
1320:
1311:
1290:
1274:
1270:
1264:
1260:
1254:
1250:
1244:
1238:
1232:
1226:
1216:
1212:
1208:
1205:
1135:
1128:
1115:
1105:matrix. The
1101:
1099:
1083:
1079:
1072:
1068:
1065:
905:
898:
878:
867:
861:
857:
842:
840:
824:
818:
815:
811:
495:
491:
487:
479:
475:
471:
465:
455:
451:
447:
440:
436:
432:
426:
420:
410:
366:
338:
333:
329:
325:
321:
317:
313:
309:
302:sparse array
301:
297:
287:
41:
3990:Hamiltonian
3914:Biadjacency
3850:Correlation
3766:Householder
3716:Commutation
3453:Vandermonde
3448:Tridiagonal
3383:Permutation
3373:Nonnegative
3358:Matrix unit
3238:Bisymmetric
3059:Binary heap
2984:Linked list
2331:: 127–137,
2205:www.anl.gov
2188:computation
2101:Sparse file
1975:ltifrontal
1934:function).
1829:into rows:
1814:, and 5 in
1607:row_start=1
1420:is located.
1398:The arrays
1140:method and
413:band matrix
407:Band matrix
358:engineering
4297:Categories
4194:algorithms
3890:Transition
3885:Stochastic
3855:Covariance
3837:statistics
3816:Symplectic
3811:Similarity
3640:Unimodular
3635:Orthogonal
3620:Involutory
3615:Invertible
3610:Projection
3606:Idempotent
3548:Convergent
3443:Triangular
3393:Polynomial
3338:Hessenberg
3308:Equivalent
3303:Elementary
3283:Copositive
3273:Conference
3233:Bidiagonal
3047:Splay tree
2951:Hash table
2832:Collection
2400:References
2211:2019-12-02
2182:2019-12-02
1626:NNZ < (
1423:The array
1295:that maps
1293:dictionary
1082:= 1, ...,
469:such that
386:compressed
373:algorithms
354:scientific
4221:CPU cache
4071:Wronskian
3995:Irregular
3985:Gell-Mann
3934:Laplacian
3929:Incidence
3909:Adjacency
3880:Precision
3845:Centering
3751:Generator
3721:Confusion
3706:Circulant
3686:Augmented
3645:Unipotent
3625:Nilpotent
3601:Congruent
3578:Stieltjes
3553:Defective
3543:Companion
3514:Redheffer
3433:Symmetric
3428:Sylvester
3403:Signature
3333:Hermitian
3313:Frobenius
3223:Arrowhead
3203:Alternant
3103:Hash tree
2989:Skip list
2936:Bit array
2837:Container
2779:694087302
2744:693784152
2661:CiteSeerX
2581:Saad 2003
2575:123079384
2526:680489638
2482:316552948
2305:Saad 2003
2286:CiteSeerX
2052:The term
2007:Armadillo
1979:assively
1886:COL_INDEX
1882:ROW_INDEX
1857:ROW_INDEX
1845:COL_INDEX
1839:COL_INDEX
1823:ROW_INDEX
1816:ROW_INDEX
1812:COL_INDEX
1611:row_end=2
1444:ROW_INDEX
1440:COL_INDEX
1425:ROW_INDEX
1414:COL_INDEX
1404:COL_INDEX
1337:(CSR) or
1131:iterative
1027:⋯
1010:⋮
1005:⋱
1000:⋮
995:⋮
983:⋯
952:⋯
875:Symmetric
871:entries.
784:⋅
774:⋅
769:⋅
764:⋅
759:⋅
751:⋅
731:⋅
726:⋅
721:⋅
698:⋅
683:⋅
675:⋅
665:⋅
655:⋅
645:⋅
637:⋅
632:⋅
622:⋅
612:⋅
599:⋅
594:⋅
579:⋅
561:⋅
556:⋅
551:⋅
546:⋅
485:whenever
4250:Software
4214:Hardware
4173:Problems
3899:Used in
3835:Used in
3796:Rotation
3771:Jacobian
3731:Distance
3711:Cofactor
3696:Carleman
3676:Adjugate
3660:Weighing
3593:inverses
3589:products
3558:Definite
3489:Identity
3479:Exchange
3472:Constant
3438:Toeplitz
3323:Hadamard
3293:Diagonal
3032:AVL tree
2911:Multiset
2860:Multimap
2847:Abstract
2755:. SIAM.
2720:. SIAM.
2640:14494429
2414:(1996).
2366:Archived
2275:(2009).
2064:See also
1953:Trilinos
1938:Software
1913:, where
837:Diagonal
369:computer
334:sparsity
18:Sparsity
4000:Overlap
3965:Density
3924:Edmonds
3801:Seifert
3761:Hessian
3726:Coxeter
3650:Unitary
3568:Hurwitz
3499:Of ones
3484:Hilbert
3418:Skyline
3363:Metzler
3353:Logical
3348:Integer
3258:Boolean
3190:classes
3086:R+ tree
3081:R* tree
3027:AA tree
2545:Bibcode
2375:6 April
2345:6412241
2048:History
1995:deal.II
1923:col_ptr
1919:row_ind
1893:logical
1810:, 8 in
1460:NNZ − 1
1370:matrix
1223:indices
1202:Storage
1102:fill-in
390:storage
324:for an
4308:Arrays
4272:LAPACK
4262:MATLAB
3919:Degree
3860:Design
3791:Random
3781:Payoff
3776:Moment
3701:Cartan
3691:Bézout
3630:Normal
3504:Pascal
3494:Lehmer
3423:Sparse
3343:Hollow
3328:Hankel
3263:Cauchy
3188:Matrix
3115:Graphs
3076:R-tree
3017:B-tree
2971:Linked
2928:Arrays
2777:
2767:
2742:
2732:
2663:
2638:
2573:
2524:
2514:
2480:
2470:
2449:
2422:
2343:
2288:
2150:
2025:ARPACK
2019:ALGLIB
1959:Eigen3
1932:sparse
1380:. Let
1066:where
883:of an
401:Banded
382:memory
310:sparse
306:matrix
4257:ATLAS
3980:Gamma
3944:Tutte
3806:Shear
3519:Shift
3509:Pauli
3458:Walsh
3368:Moore
3248:Block
3009:Trees
2882:Queue
2877:Stack
2825:Types
2636:S2CID
2571:S2CID
2495:(PDF)
2369:(PDF)
2362:(PDF)
2341:S2CID
2321:(PDF)
2282:(PDF)
2118:Notes
2031:SLEPc
2013:SciPy
1969:MUMPS
1947:PETSc
1837:(and
1798:4 × 6
1796:is a
1585:4 × 4
1583:is a
1301:pairs
1142:GMRES
1129:Both
851:as a
490:<
450:>
314:dense
304:is a
4236:SIMD
3786:Pick
3756:Gram
3524:Zero
3228:Band
3098:Trie
3054:Heap
2872:List
2775:OCLC
2765:ISBN
2740:OCLC
2730:ISBN
2522:OCLC
2512:ISBN
2478:OCLC
2468:ISBN
2447:ISBN
2420:ISBN
2377:2019
2256:See
2242:See
2228:See
2148:ISBN
2001:DUNE
1891:For
1884:and
1617:and
1615:V =
1609:and
1438:and
1402:and
1333:The
1230:and
1100:The
375:and
348:and
296:, a
292:and
4226:TLB
3875:Hat
3608:or
3591:or
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