Welcome, guest | Sign In | My Account | Store | Cart

'sparse' is a matrix class based on a dictionary to store data using 2-element tuples (i,j) as keys (i is the row and j the column index). The common matrix operations such as 'dot' for the inner product, multiplication/division by a scalar, indexing/slicing, etc. are overloaded for convenience. When used in conjunction with the 'vector' class, 'dot' products also apply between matrics and vectors. Two methods, 'CGsolve' and 'biCGsolve', are provided to solve linear systems. Tested using Python 2.2.

Python, 420 lines
  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
#!/usr/bin/env python

import vector
import math, types, operator

"""
A sparse matrix class based on a dictionary, supporting matrix (dot)
product and a conjugate gradient solver. 

In this version, the sparse class inherits from the dictionary; this
requires Python 2.2 or later.
"""

class sparse(dict):
	"""
	A complex sparse matrix 
	A. Pletzer 5 Jan 00/12 April 2002

	Dictionary storage format { (i,j): value, ... }
	where (i,j) are the matrix indices
       	"""

	# no c'tor

	def size(self):
		" returns # of rows and columns "
		nrow = 0
		ncol = 0
		for key in self.keys():
			nrow = max([nrow, key[0]+1])
			ncol = max([ncol, key[1]+1])
		return (nrow, ncol)

	def __add__(self, other):
		res = sparse(self.copy())
		for ij in other:
			res[ij] = self.get(ij,0.) + other[ij]
		return res
		
	def __neg__(self):
		return sparse(zip(self.keys(), map(operator.neg, self.values())))

	def __sub__(self, other):
		res = sparse(self.copy())
		for ij in other:
			res[ij] = self.get(ij,0.) - other[ij]
		return res
		
	def __mul__(self, other):
		" element by element multiplication: other can be scalar or sparse "
		try:
			# other is sparse
			nval = len(other)
			res = sparse()
			if nval < len(self):
				for ij in other:
					res[ij] = self.get(ij,0.)*other[ij]
			else:
				for ij in self:
					res[ij] = self[ij]*other.get(ij,0j)
			return res
		except:
			# other is scalar
			return sparse(zip(self.keys(), map(lambda x: x*other, self.values())))


	def __rmul__(self, other): return self.__mul__(other)

	def __div__(self, other):
		" element by element division self/other: other is scalar"
		return sparse(zip(self.keys(), map(lambda x: x/other, self.values())))
		
	def __rdiv__(self, other):
		" element by element division other/self: other is scalar"
		return sparse(zip(self.keys(), map(lambda x: other/x, self.values())))

	def abs(self):
		return sparse(zip(self.keys(), map(operator.abs, self.values())))

	def out(self):
		print '# (i, j) -- value'
		for k in self.keys():
			print k, self[k]

	def plot(self, width_in=400, height_in=400):

		import colormap
		import Tkinter

		cmax = max(self.values())
		cmin = min(self.values())
		
		offset =  0.05*min(width_in, height_in)
		xmin, ymin, xmax, ymax = 0,0,self.size()[0], self.size()[1]
		scale =  min(0.9*width_in, 0.9*height_in)/max(xmax-xmin, ymax-ymin)

		root = Tkinter.Tk()
		frame = Tkinter.Frame(root)
		frame.pack()
		
		text = Tkinter.Label(width=20, height=10, text='matrix sparsity')
		text.pack()
		

		canvas = Tkinter.Canvas(bg="black", width=width_in, height=height_in)
		canvas.pack()

		button = Tkinter.Button(frame, text="OK?", fg="red", command=frame.quit)
		button.pack()

		for index in self.keys():
			ix, iy = index[0], ymax-index[1]-1
			ya, xa = offset+scale*(ix  ), height_in -offset-scale*(iy  )
			yb, xb = offset+scale*(ix+1), height_in -offset-scale*(iy  )
			yc, xc = offset+scale*(ix+1), height_in -offset-scale*(iy+1)
			yd, xd = offset+scale*(ix  ), height_in -offset-scale*(iy+1)
			color = colormap.strRgb(self[index], cmin, cmax)
			canvas.create_polygon(xa, ya, xb, yb, xc, yc, xd, yd, fill=color)
		
		root.mainloop()

	def CGsolve(self, x0, b, tol=1.0e-10, nmax = 1000, verbose=1):
		"""
		Solve self*x = b and return x using the conjugate gradient method
		"""
		if not vector.isVector(b):
			raise TypeError, self.__class__,' in solve '
		else:
			if self.size()[0] != len(b) or self.size()[1] != len(b):
				print '**Incompatible sizes in solve'
				print '**size()=', self.size()[0], self.size()[1]
				print '**len=', len(b)
			else:
				kvec = diag(self) # preconditionner
				n = len(b)
				x = x0 # initial guess
				r =  b - dot(self, x)
				try:
					w = r/kvec
				except: print '***singular kvec'
				p = vector.zeros(n);
				beta = 0.0;
				rho = vector.dot(r, w);
				err = vector.norm(dot(self,x) - b);
				k = 0
				if verbose: print " conjugate gradient convergence (log error)"
				while abs(err) > tol and k < nmax:
					p = w + beta*p;
					z = dot(self, p);
					alpha = rho/vector.dot(p, z);
					r = r - alpha*z;
					w = r/kvec;
					rhoold = rho;
					rho = vector.dot(r, w);
					x = x + alpha*p;
					beta = rho/rhoold;
					err = vector.norm(dot(self, x) - b);
					if verbose: print k,' %5.1f ' % math.log10(err)
					k = k+1
				return x
				
		
	    		
	def biCGsolve(self,x0, b, tol=1.0e-10, nmax = 1000):
		
		"""
		Solve self*x = b and return x using the bi-conjugate gradient method
		"""

		try:
			if not vector.isVector(b):
				raise TypeError, self.__class__,' in solve '
			else:
				if self.size()[0] != len(b) or self.size()[1] != len(b):
					print '**Incompatible sizes in solve'
					print '**size()=', self.size()[0], self.size()[1]
					print '**len=', len(b)
				else:
					kvec = diag(self) # preconditionner 
					n = len(b)
					x = x0 # initial guess
					r =  b - dot(self, x)
					rbar =  r
					w = r/kvec;
					wbar = rbar/kvec;
					p = vector.zeros(n);
					pbar = vector.zeros(n);
					beta = 0.0;
					rho = vector.dot(rbar, w);
					err = vector.norm(dot(self,x) - b);
					k = 0
					print " bi-conjugate gradient convergence (log error)"
					while abs(err) > tol and k < nmax:
						p = w + beta*p;
						pbar = wbar + beta*pbar;
						z = dot(self, p);
						alpha = rho/vector.dot(pbar, z);
						r = r - alpha*z;
						rbar = rbar - alpha* dot(pbar, self);
						w = r/kvec;
						wbar = rbar/kvec;
						rhoold = rho;
						rho = vector.dot(rbar, w);
						x = x + alpha*p;
						beta = rho/rhoold;
						err = vector.norm(dot(self, x) - b);
						print k,' %5.1f ' % math.log10(err)
						k = k+1
					return x
			
		except: print 'ERROR ',self.__class__,'::biCGsolve'


	def save(self, filename, OneBased=0):
		"""
		Save matrix in file <filaname> using format:
		OneBased, nrow, ncol, nnonzeros
		[ii, jj, data]

		"""
		m = n = 0
		nnz = len(self)
		for ij in self.keys():
			m = max(ij[0], m)
			n = max(ij[1], n)

		f = open(filename,'w')
		f.write('%d %d %d %d\n' % (OneBased, m+1,n+1,nnz))
		for ij in self.keys():
			i,j = ij
			f.write('%d %d %20.17f \n'% \
				(i+OneBased,j+OneBased,self[ij]))
		f.close()
				
###############################################################################

def isSparse(x):
    return hasattr(x,'__class__') and x.__class__ is sparse

def transp(a):
	" transpose "
	new = sparse({})
	for ij in a:
		new[(ij[1], ij[0])] = a[ij]
	return new

def dotDot(y,a,x):
	" double dot product y^+ *A*x "
	if vector.isVector(y) and isSparse(a) and vector.isVector(x):
		res = 0.
		for ij in a.keys():
			i,j = ij
			res += y[i]*a[ij]*x[j]
		return res
	else:
		print 'sparse::Error: dotDot takes vector, sparse , vector as args'

def dot(a, b):
	" vector-matrix, matrix-vector or matrix-matrix product "
	if isSparse(a) and vector.isVector(b):
		new = vector.zeros(a.size()[0])
		for ij in a.keys():
			new[ij[0]] += a[ij]* b[ij[1]]
		return new
	elif vector.isVector(a) and isSparse(b):
		new = vector.zeros(b.size()[1])
		for ij in b.keys():
			new[ij[1]] += a[ij[0]]* b[ij]
		return new
	elif isSparse(a) and isSparse(b):
		if a.size()[1] != b.size()[0]:
			print '**Warning shapes do not match in dot(sparse, sparse)'
		new = sparse({})
		n = min([a.size()[1], b.size()[0]])
		for i in range(a.size()[0]):
			for j in range(b.size()[1]):
				sum = 0.
				for k in range(n):
					sum += a.get((i,k),0.)*b.get((k,j),0.)
				if sum != 0.:
					new[(i,j)] = sum
		return new
	else:
		raise TypeError, 'in dot'

def diag(b):
	# given a sparse matrix b return its diagonal
	res = vector.zeros(b.size()[0])
	for i in range(b.size()[0]):
		res[i] = b.get((i,i), 0.)
	return res
		
def identity(n):
	if type(n) != types.IntType:
		raise TypeError, ' in identity: # must be integer'
	else:
		new = sparse({})
		for i in range(n):
			new[(i,i)] = 1+0.
		return new

###############################################################################
if __name__ == "__main__":

	print 'a = sparse()'
	a = sparse()

	print 'a.__doc__=',a.__doc__

	print 'a[(0,0)] = 1.0'
	a[(0,0)] = 1.0
	a.out()

	print 'a[(2,3)] = 3.0'
	a[(2,3)] = 3.0
	a.out()

	print 'len(a)=',len(a)
	print 'a.size()=', a.size()
			
	b = sparse({(0,0):2.0, (0,1):1.0, (1,0):1.0, (1,1):2.0, (1,2):1.0, (2,1):1.0, (2,2):2.0})
	print 'a=', a
	print 'b=', b
	b.out()

	print 'a+b'
	c = a + b
	c.out()

	print '-a'
	c = -a
	c.out()
	a.out()

	print 'a-b'
	c = a - b
	c.out()

	print 'a*1.2'
	c = a*1.2
	c.out()


	print '1.2*a'
	c = 1.2*a
	c.out()
	print 'a=', a

	print 'dot(a, b)'
	print 'a.size()[1]=',a.size()[1],' b.size()[0]=', b.size()[0]
	c = dot(a, b)
	c.out()

	print 'dot(b, a)'
	print 'b.size()[1]=',b.size()[1],' a.size()[0]=', a.size()[0]
	c = dot(b, a)
	c.out()

	try:
		print 'dot(b, vector.vector([1,2,3]))'
		c = dot(b, vector.vector([1,2,3]))
		c.out()
	
		print 'dot(vector.vector([1,2,3]), b)'
		c = dot(vector.vector([1,2,3]), b)
		c.out()

		print 'b.size()=', b.size()
	except: pass
	
	print 'a*b -> element by element product'
	c = a*b
	c.out()

	print 'b*a -> element by element product'
	c = b*a
	c.out()
	
	print 'a/1.2'
	c = a/1.2
	c.out()

	print 'c = identity(4)'
	c = identity(4)
	c.out()

	print 'c = transp(a)'
	c = transp(a)
	c.out()


	b[(2,2)]=-10.0
	b[(2,0)]=+10.0

	try:
		import vector
		print 'Check conjugate gradient solver'
		s = vector.vector([1, 0, 0])
		print 's'
		s.out()
		x0 = s 
		print 'x = b.biCGsolve(x0, s, 1.0e-10, len(b)+1)'
		x = b.biCGsolve(x0, s, 1.0e-10,  len(b)+1)
		x.out()

		print 'check validity of CG'
		c = dot(b, x) - s
		c.out()
	except: pass

	print 'plot b matrix'
	b.out()
	b.plot()

	print 'del b[(2,2)]'
	del b[(2,2)]

	print 'del a'
	del a
	#a.out()

I'm using this class in the context of a two-dimensional finite element code. When discretized, the partial differential equation reduces to a sparse matrix system. Because 'sparse' stores the data in a dictionary, the size of the problem need not be known before hand (the sparse matrix elements are filled up along the way). For such problems, the resulting matrix tends to be extremely sparse so that there is little advantage in storing data contiguously in memory; the dictionary random based storage turns out to be appropriate.

'sparse' in conjunction with 'vector' (also available as a Python recipe) supports many matrix-vector operations (+, dot product etc) as well as elementwise operations (sin, cos, ...). Thus, in order to use 'sparse' you will need to download 'vector'. 'sparse' comes in addition with a method for solving linear matrix systems based on the conjugate gradient method.

If you want a picture of your matrix using Tkinter, I suggest that you also download 'colormap'.

Just type in 'python sparse.py' to test some of sparse's functionality.

PS This version uses Python 2.2's new feature for deriving a class from a dictionary type. It also runs significantly faster than the previously posted version, thanks to the use of map/reduce and lambda.

--Alex.

5 comments

Brett Morgan 20 years, 2 months ago  # | flag

Uhh Source Code? Where'd the code go? -- Brett Morgan

Pierre Johnson 19 years, 4 months ago  # | flag

great work! This is great stuff!

Just a little note. Line that reads import sparse isn't needed.

Anand 18 years, 2 months ago  # | flag

Cool stuff. This is really cool stuff. I suggest that it

be made into a python module. Probably too big

for the Cookbook... ;-)

-Anand

Frank Horowitz 17 years, 9 months ago  # | flag

Typo? I think there's a typo in __mul__; the res = csparse() should be res = sparse(), no?

Alexander Pletzer (author) 16 years, 7 months ago  # | flag

You're right. Thanks for spotting this typo. --Alex.