def solve_qr_b_givens(a, b):
gMatrixArray = qr_fact_givens.qr_fact_givens(a)
q = numpy.asmatrix(gMatrixArray[0])
r = numpy.asmatrix(gMatrixArray[1])
d = common.mult_mat(numpy.transpose(q), b)
rSize = r.shape[0]
x = "0"
for i in range(0, (rSize - 1)):
x += "; 0"
x = numpy.matrix(x, dtype=float)
# back substitution, i = row num, j = col num
# r = upper triangular matrix, so it starts at bottom right corner and works its way up
# gets each x value, from x4 to x1
# [x1+x2+x3+x4]
# [ x2+x3+x4]
# [ x3+x4]
# [ x4]
for i in range(rSize - 1, -1, -1):
for j in range(rSize - 1, -1, -1):
if j >= i:
# if not at position pertaining to current x (i.e. a diagonal entry)
if i != j:
d[i] = d[i] - (r[i, j] * d[j])
elif i == j:
x[i] = d[i] / r[i, j]
return (x, common.error(common.mult_mat(a, x), b))
def solve_33_iter(B, AsubB, x0, b, tol, max_iter):
"""Iteratively solve Ax = b for a 3x3 system of equations, within a given tolerance using a given decomposition of A.
Args:
B (matrix): 3x3 primary part of A.
AsubB (matrix): 3x3 secondary part of A.
x0 (matrix): 3x1 starting vector.
b (matrix): 3x1 b for the system of equations.
tol (float): Tolerance for the approximate solution.
max_iter (int): Maximum number of attempted iterations.
Returns:
tuple(matrix, int): 3x1 approximate solution of system and number of iterations performed;
Raises:
ValueError: If any inputs are not expected type.
DimensionError: If A or AsubB are not 3x3 or x0 or b are not 3x1.
ConvergenceError: If system does not converge on a solution before max_iter iterations.
"""
if (type(B) is not numpy.matrixlib.defmatrix.matrix) or (type(AsubB) is not numpy.matrixlib.defmatrix.matrix) or (type(x0) is not numpy.matrixlib.defmatrix.matrix) or (type(b) is not numpy.matrixlib.defmatrix.matrix) or (type(tol) is not float) or (type(max_iter) is not int):
raise TypeError('Expected matrix, matrix, matrix, matrix, float, and int, got ' + str(type(B)) + ', ' + str(type(AsubB)) + ', ' + str(type(x0)) + ', ' + str(type(b)) + ', ' + str(type(tol)), ', and ' + str(type(max_iter)))
if B.shape != (3, 3):
raise common.DimensionError('Expected 3x3 matrix for B, got ' + str(B.shape))
if AsubB.shape != (3, 3):
raise common.DimensionError('Expected 3x3 matrix for AsubB, got ' + str(AsubB.shape))
if x0.shape != (3, 1):
raise common.DimensionError('Expected 3x1 matrix for x0, got ' + str(x0.shape))
if b.shape != (3, 1):
raise common.DimensionError('Expected 3x1 matrix for b, got ' + str(b.shape))
Bi = common.inv_33_lower_tri(B)
curx = x0
itr = 0;
while itr < max_iter:
nextx = common.mult_mat(Bi, (b - common.mult_mat(AsubB, curx)))
if common.error(nextx, curx) <= tol:
return (nextx, itr)
curx = nextx
itr += 1
raise common.ConvergenceError("System did not converge before " + str(max_iter) + " iterations")
def solve_lu_b(a, b) : matrixArray = lu_fact.lu_fact(a) l = numpy.asmatrix(matrixArray[0]) u = numpy.asmatrix(matrixArray[1]) d = numpy.matrix(b, dtype=float) size = u.shape[0] y = "0" for i in range(0,(size-1)) : y += "; 0" y = numpy.matrix(y, dtype = float) for i in range(0, size) : for j in range(0, size) : if (j <= i) : if (i != j) : if ((l[i,j] * d[j]) < 0) : d[i] = d[i] + (0 - (l[i,j] * d[j])) elif ((l[i,j] * d[j]) > 0) : d[i] = d[i] - (l[i,j] * d[j]) elif (i == j) : if l[i,j] != 0 : y[i] = d[i] / l[i,j] else : y[i] = d[i] x = "0" for i in range(0,(size-1)) : x += "; 0" x = numpy.matrix(x, dtype = float) for i in range(size - 1, -1, -1) : for j in range(size - 1, -1, -1) : if (j >= i) : if (i != j) : if ( (u[i,j] * y[j]) < 0) : y[i] = y[i] + (0 - (u[i,j] * y[j])) elif ((u[i,j] * y[j]) > 0) : y[i] = y[i] - (u[i,j] * y[j]) elif (i == j) : if l[i,j] != 0 : x[i] = y[i] / u[i,j] else : x[i] = y[i] return (x, common.error(common.mult_mat(a, x), b))
def lu_fact(matrix):
matrixCopy = numpy.matrix(matrix, dtype=float);
columns = matrix.shape[1]
rows = matrix.shape[0]
identity = matlib.identity(rows);
#for each non-trivial pivot
for m in range (0, rows-1):
#for each row under pivot
for n in range (m+1, rows):
#Subtract leading/pivot times pivot row from current row
subRow = matrixCopy[m,:]/float(matrixCopy[m,m])*matrixCopy[m,n]
matrixCopy[n] = matrixCopy[n] - subRow
#Put leading/pivot in corresponding location in identity
identity[n,m] = matrixCopy[m,n]/float(matrixCopy[m,m])
#return L, U, and error
return (identity, matrixCopy, common.error(common.mult_mat(identity, matrixCopy), matrix))
def qr_fact_househ(matrix): #sets up various things that are necessary uSetUp = "" size = matrix.shape colSize = size[0] rowSize = size[1] cont = True i = 0 normalize = 0 houseHolder = matrix tempHouseHolder = houseHolder count = 0 #nested for loop that creates uSetUp #uSetUp = contains string of numbers (end of each row signified with ;) that will make u for i in range(0, colSize) : count = 1 uSetUp = "" normalize = 0 for j in range(i, rowSize) : if j != i : if uSetUp != "" : uSetUp += ";" uSetUp += "%f" % (houseHolder[j,i]) #added to u1 to get full u normalize += houseHolder[j,i] ** 2 #only "counts" if houseHolder[i,j] is nonzero if houseHolder[j,i] != 0 : count += 1 #if "count is greater than 1" if count > 1 : #gets normalized x and adds it to u1 and prepares to add it to the string of numbers afterNormalize = math.sqrt(normalize + (houseHolder[i,i]**2)) normalizedU = houseHolder[i,i] + afterNormalize newValStore = "%f;" % (normalizedU) #actually adds it to the string of numbers uSetUp = newValStore + uSetUp #fairly obvious u = numpy.matrix(uSetUp) uT = numpy.transpose(u) identMatrix = numpy.identity(u.shape[0]) #creates (2uuT)/(magnitude(u)^2) numerator = numpy.multiply(2,common.mult_mat(u, uT)) normalize += (houseHolder[i,i] + afterNormalize) ** 2 numerator = numpy.divide(numerator, normalize) #creates H(n) tempHouseHolder = identMatrix - numerator #resizes various matrices to make it easier in the long run numColSize = numerator.shape[0] numRowSize = numerator.shape[1] rowSizeDiff = rowSize - numRowSize colSizeDiff = colSize - numColSize temp = tempHouseHolder tempHouseHolder = numpy.identity(colSize) for i in range(0, colSize) : for j in range(0, rowSize) : if j >= (rowSizeDiff) and i >= (colSizeDiff) : tempHouseHolder[i,j] = temp[i - rowSizeDiff, j - colSizeDiff] #gets new H from H multiplied H(n) or original matrix if (houseHolder == matrix).all() : q = numpy.transpose(tempHouseHolder) houseHolder = common.mult_mat(numpy.matrixlib.defmatrix.matrix(tempHouseHolder), matrix) else : q = common.mult_mat(numpy.matrixlib.defmatrix.matrix(q),numpy.transpose(numpy.matrixlib.defmatrix.matrix(tempHouseHolder))) houseHolder = common.mult_mat(numpy.matrixlib.defmatrix.matrix(tempHouseHolder), numpy.matrixlib.defmatrix.matrix(houseHolder)) error = common.error(common.mult_mat(numpy.matrixlib.defmatrix.matrix(q),numpy.matrixlib.defmatrix.matrix(houseHolder)),matrix) return (q, houseHolder, error)
