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* [42]One Dimensional Minimization
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* [44]Multidimensional Minimization
* [45]Linear Least-Squares Fitting
* [46]Nonlinear Least-Squares Fitting
* [47]Basis Splines
* [48]Sparse Matrices
+ [49]Data types
+ [50]Sparse Matrix Storage Formats
o [51]Coordinate Storage (COO)
o [52]Compressed Sparse Column (CSC)
o [53]Compressed Sparse Row (CSR)
+ [54]Overview
+ [55]Allocation
+ [56]Accessing Matrix Elements
+ [57]Initializing Matrix Elements
+ [58]Reading and Writing Matrices
+ [59]Copying Matrices
+ [60]Exchanging Rows and Columns
+ [61]Matrix Operations
+ [62]Matrix Properties
+ [63]Finding Maximum and Minimum Elements
+ [64]Compressed Format
+ [65]Conversion Between Sparse and Dense Matrices
+ [66]Examples
+ [67]References and Further Reading
* [68]Sparse BLAS Support
* [69]Sparse Linear Algebra
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* Sparse Matrices
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____________________________________________________________________________
Sparse Matrices[82]¶
This chapter describes functions for the construction and manipulation of sparse
matrices, matrices which are populated primarily with zeros and contain only a
few non-zero elements. Sparse matrices often appear in the solution of partial
differential equations. It is beneficial to use specialized data structures and
algorithms for storing and working with sparse matrices, since dense matrix
algorithms and structures can be prohibitively slow and use huge amounts of
memory when applied to sparse matrices.
The header file gsl_spmatrix.h contains the prototypes for the sparse matrix
functions and related declarations.
Data types[83]¶
All the functions are available for each of the standard data-types. The versions
for double have the prefix gsl_spmatrix, Similarly the versions for
single-precision float arrays have the prefix gsl_spmatrix_float. The full list
of available types is given below,
Prefix
Type
gsl_spmatrix
double
gsl_spmatrix_float
float
gsl_spmatrix_long_double
long double
gsl_spmatrix_int
int
gsl_spmatrix_uint
unsigned int
gsl_spmatrix_long
long
gsl_spmatrix_ulong
unsigned long
gsl_spmatrix_short
short
gsl_spmatrix_ushort
unsigned short
gsl_spmatrix_char
char
gsl_spmatrix_uchar
unsigned char
gsl_spmatrix_complex
complex double
gsl_spmatrix_complex_float
complex float
gsl_spmatrix_complex_long_double
complex long double
Sparse Matrix Storage Formats[84]¶
GSL currently supports three storage formats for sparse matrices: the coordinate
(COO) representation, compressed sparse column (CSC) and compressed sparse row
(CSR) formats. These are discussed in more detail below. In order to illustrate
the different storage formats, the following sections will reference this M -by-
N sparse matrix, with M=4 and N=5 :
\begin{pmatrix} 9 & 0 & 0 & 0 & -3 \\ 4 & 7 & 0 & 0 & 0 \\ 0 & 8 & -1 & 8 & 0 \\
4 & 0 & 5 & 6 & 0 \end{pmatrix}
The number of non-zero elements in the matrix, also abbreviated as nnz is equal
to 10 in this case.
Coordinate Storage (COO)[85]¶
The coordinate storage format, also known as triplet format, stores triplets
(i,j,x) for each non-zero element of the matrix. This notation means that the
(i,j) element of the matrix A is A_{ij} = x . The matrix is stored using three
arrays of the same length, representing the row indices, column indices, and
matrix data. For the reference matrix above, one possible storage format is:
data
9
7
4
8
-3
-1
8
5
6
4
row
0
1
1
2
0
2
2
3
3
3
col
0
1
0
1
4
2
3
2
3
0
Note that this representation is not unique - the coordinate triplets may appear
in any ordering and would still represent the same sparse matrix. The length of
the three arrays is equal to the number of non-zero elements in the matrix, nnz,
which in this case is 10 . The coordinate format is extremely convenient for
sparse matrix assembly, the process of adding new elements, or changing existing
elements, in a sparse matrix. However, it is generally not suitable for the
efficient implementation of matrix-matrix products, or matrix factorization
algorithms. For these applications it is better to use one of the compressed
formats discussed below.
In order to faciliate efficient sparse matrix assembly, GSL stores the coordinate
data in a balanced binary search tree, specifically an AVL tree, in addition to
the three arrays discussed above. This allows GSL to efficiently determine
whether an entry (i,j) already exists in the matrix, and to replace an existing
matrix entry with a new value, without needing to search unsorted arrays.
Compressed Sparse Column (CSC)[86]¶
Compressed sparse column storage stores each column of non-zero values in the
sparse matrix in a continuous memory block, keeping pointers to the beginning of
each column in that memory block, and storing the row indices of each non-zero
element. For the reference matrix above, these arrays look like
data
9
4
4
7
8
-1
5
8
6
-3
row
0
1
3
1
2
2
3
2
3
0
col_ptr
0
3
5
7
9
10
The data and row arrays are of length nnz and are the same as the COO storage
format. The col_ptr array has length N+1 , and col_ptr[j] gives the index in data
of the start of column j. Therefore, the j -th column of the matrix is stored in
data[col_ptr[j]], data[col_ptr[j] + 1], ..., data[col_ptr[j+1] - 1]. The last
element of col_ptr is nnz.
Compressed Sparse Row (CSR)[87]¶
Compressed row storage stores each row of non-zero values in a continuous memory
block, keeping pointers to the beginning of each row in the block and storing the
column indices of each non-zero element. For the reference matrix above, these
arrays look like
data
9
-3
4
7
8
-1
8
4
5
6
col
0
4
0
1
1
2
3
0
2
3
row_ptr
0
2
4
7
10
The data and col arrays are of length nnz and are the same as the COO storage
format. The row_ptr array has length M+1 , and row_ptr[i] gives the index in data
of the start of row i. Therefore, the i -th row of the matrix is stored in
data[row_ptr[i]], data[row_ptr[i] + 1], ..., data[row_ptr[i+1] - 1]. The last
element of row_ptr is nnz.
Overview[88]¶
These routines provide support for constructing and manipulating sparse matrices
in GSL, using an API similar to [89]gsl_matrix. The basic structure is called
[90]gsl_spmatrix.
type gsl_spmatrix[91]¶
This structure is defined as:
typedef struct
{
size_t size1;
size_t size2;
int *i;
double *data;
int *p;
size_t nzmax;
size_t nz;
[ ... variables for binary tree and memory management ... ]
size_t sptype;
} gsl_spmatrix;
This defines a size1-by-size2 sparse matrix. The number of non-zero
elements currently in the matrix is given by nz. For the triplet
representation, i, p, and data are arrays of size nz which contain the row
indices, column indices, and element value, respectively. So if data[k] =
A(i,j) , then i = i[k] and j = p[k] .
For compressed column storage, i and data are arrays of size nz containing
the row indices and element values, identical to the triplet case. p is an
array of size size2 + 1 where p[j] points to the index in data of the
start of column j. Thus, if data[k] = A(i,j) , then i = i[k] and p[j] i[i]);
printf("]\n");
printf("p = [ ");
for (i = 0; i < B->size2 + 1; ++i)
printf("%d, ", B->p[i]);
printf("]\n");
printf("d = [ ");
for (i = 0; i < B->nz; ++i)
printf("%g, ", B->data[i]);
printf("]\n");
/* convert to compressed row format */
C = gsl_spmatrix_crs(A);
printf("matrix in compressed row format:\n");
printf("i = [ ");
for (i = 0; i < C->nz; ++i)
printf("%d, ", C->i[i]);
printf("]\n");
printf("p = [ ");
for (i = 0; i < C->size1 + 1; ++i)
printf("%d, ", C->p[i]);
printf("]\n");
printf("d = [ ");
for (i = 0; i < C->nz; ++i)
printf("%g, ", C->data[i]);
printf("]\n");
gsl_spmatrix_free(A);
gsl_spmatrix_free(B);
gsl_spmatrix_free(C);
return 0;
}
References and Further Reading[359]¶
The algorithms used by these functions are described in the following sources,
* Davis, T. A., Direct Methods for Sparse Linear Systems, SIAM, 2006.
* CSparse software library,
[360]https://www.cise.ufl.edu/research/sparse/CSparse
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160. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
161. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_set_zero
162. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_set_zero
163. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
164. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
165. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
166. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#reading-and-writing-matrices
167. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
168. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fwrite
169. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fwrite
170. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fwrite
171. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
172. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
173. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
174. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
175. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fread
176. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fread
177. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fread
178. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fread
179. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
180. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
181. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
182. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
183. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fprintf
184. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fprintf
185. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fprintf
186. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fprintf
187. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fprintf
188. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
189. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
190. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
191. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
192. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fscanf
193. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_fscanf
194. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
195. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#copying-matrices
196. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
197. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
198. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_memcpy
199. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_memcpy
200. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_memcpy
201. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
202. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
203. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
204. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#exchanging-rows-and-columns
205. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
206. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
207. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose_memcpy
208. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose_memcpy
209. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose_memcpy
210. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose_memcpy
211. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose_memcpy
212. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
213. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
214. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
215. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
216. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose
217. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_transpose
218. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
219. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
220. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
221. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#matrix-operations
222. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
223. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale
224. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale
225. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale
226. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale
227. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
228. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
229. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
230. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
231. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
232. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_columns
233. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_columns
234. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_columns
235. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_columns
236. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
237. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
238. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
239. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
240. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector
241. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_rows
242. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_rows
243. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_rows
244. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_scale_rows
245. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
246. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
247. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
248. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
249. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
250. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
251. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_add
252. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
253. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
254. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
255. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
256. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_add
257. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_add
258. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_add
259. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_add
260. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_add
261. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
262. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
263. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
264. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
265. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
266. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_sub
267. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_sub
268. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_sub
269. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_sub
270. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_dense_sub
271. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
272. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
273. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
274. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#matrix-properties
275. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
276. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_type
277. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_type
278. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
279. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
280. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
281. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
282. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_nnz
283. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_nnz
284. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
285. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
286. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
287. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
288. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
289. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_equal
290. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_equal
291. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_equal
292. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_equal
293. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_equal
294. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
295. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
296. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
297. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
298. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_norm1
299. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_norm1
300. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
301. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
302. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
303. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#finding-maximum-and-minimum-elements
304. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
305. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_minmax
306. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_minmax
307. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_minmax
308. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_minmax
309. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
310. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
311. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
312. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
313. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_min_index
314. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_min_index
315. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_min_index
316. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_min_index
317. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
318. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
319. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
320. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#compressed-format
321. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
322. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
323. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_csc
324. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
325. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_csc
326. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_csc
327. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
328. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
329. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
330. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_csr
331. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
332. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_csr
333. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_csr
334. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
335. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
336. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
337. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_compress
338. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_compress
339. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_compress
340. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_compress
341. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#conversion-between-sparse-and-dense-matrices
342. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
343. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
344. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
345. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
346. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_d2sp
347. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_d2sp
348. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_d2sp
349. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
350. http://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_matrix
351. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix
352. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_sp2d
353. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_sp2d
354. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#c.gsl_spmatrix_sp2d
355. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-coo
356. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csc
357. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#sec-spmatrix-csr
358. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#examples
359. http://www.gnu.org/software/gsl/doc/html/spmatrix.html#references-and-further-reading
360. https://www.cise.ufl.edu/research/sparse/CSparse
361. http://www.gnu.org/software/gsl/doc/html/spblas.html
362. http://www.gnu.org/software/gsl/doc/html/bspline.html
363. https://www.sphinx-doc.org/
364. https://github.com/readthedocs/sphinx_rtd_theme
365. https://readthedocs.org/
Hidden links:
367. http://www.gnu.org/software/gsl/doc/html/index.html
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