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'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
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#!/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 22 years, 6 months ago  # | flag

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

Pierre Johnson 21 years, 8 months ago  # | flag

great work! This is great stuff!

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

Anand 20 years, 6 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 20 years, 1 month ago  # | flag

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

Alexander Pletzer (author) 18 years, 11 months ago  # | flag

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