# fixed_dof_solve

## `fixed_dof_solve_precompute`

Source code in `src/gpytoolbox/fixed_dof_solve.py`
 ``` 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``` ``````class fixed_dof_solve_precompute: def __init__(self, A, k=None): """Prepare a precomputation object to efficiently solve a linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution. For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows. This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values. Parameters ---------- A : (n,n) scipy csc matrix square matrix for the linear system k : (o,) numpy int array index vector of fixed degrees of freedom Returns ------- precomputed : precomputation object that can be used to solve the problem See Also -------- min_quad_with_fixed Examples -------- ```python >>> import gpytoolbox as gpy >>> import numpy as np >>> import scipy as sp >>> >>> # This matrix is not symmetric! >>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]])) >>> k = np.array([2]) >>> precomp = gpy.fixed_dof_solve_precompute(A, k=k) >>> >>> b = np.array([0.,0.,0.]) >>> y = np.array([5.]) >>> u = precomp.solve(b=b, y=y) >>> u array([-5., 0., 5.]) >>> >>> # A*u matches b, except for indices where constraints apply. >>> A*u - b array([0., 0., 5.]) ``` """ self.n = A.shape[0] assert A.shape[1] == self.n assert self.n>0 # self.A = A.copy() if k is None: self.o = 0 self.k = None else: self.o = k.shape[0] assert k.shape == (self.o,) assert np.min(k)>=0 and np.max(k)>> import gpytoolbox as gpy >>> import numpy as np >>> import scipy as sp >>> >>> # This matrix is not symmetric! >>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]])) >>> k = np.array([2]) >>> precomp = gpy.fixed_dof_solve_precompute(A, k=k) >>> >>> b = np.array([0.,0.,0.]) >>> y = np.array([5.]) >>> u = precomp.solve(b=b, y=y) >>> u array([-5., 0., 5.]) >>> >>> # A*u matches b, except for indices where constraints apply. >>> A*u - b array([0., 0., 5.]) ``` """ def cp(x): if x is None or np.isscalar(x): return 0 if len(x.shape)==1: return 0 return x.shape[1] p = max([cp(b), cp(y)]) assert b is None or np.isscalar(b) or (p==0 and b.shape==(self.n,)) or (p>0 and b.shape==(self.n,p)) assert y is None or (np.isscalar(y) and self.o>0) or (p==0 and y.shape==(self.o,)) or (p>0 and y.shape==(self.o,p)) # Get everything to full dimensions if b is None: b = 0. if np.isscalar(b): if p==0: b = np.full(self.n, b) else: b = np.full((self.n,p), b) if y is None and self.o>0: y = 0. if np.isscalar(y) and self.o>0: if p==0: y = np.full(self.o, y) else: y = np.full((self.o,p), y) # Modified rhs based on known values if self.o==0: rhs = b else: b_reduced = b[self.ki] if p==0 else b[self.ki,:] rhs = b_reduced - self.A_for_extra_b*y ured = self.solver(rhs) if self.o==0: u = ured else: if p==0: u = np.empty(self.n, dtype=np.float64) u[self.ki] = ured u[self.k] = y else: u = np.empty((self.n,p), dtype=np.float64) u[self.ki,:] = ured u[self.k,:] = y return u ``````

### `__init__(A, k=None)`

Prepare a precomputation object to efficiently solve a linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution.

For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows.

This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values.

Parameters:

Name Type Description Default
`A` `(n,n) scipy csc matrix`

square matrix for the linear system

required
`k` `(o,) numpy int array`

index vector of fixed degrees of freedom

`None`

Returns:

Name Type Description
`precomputed` `precomputation object that can be used to solve the problem`

Examples:

``````>>> import gpytoolbox as gpy
>>> import numpy as np
>>> import scipy as sp
>>>
>>> # This matrix is not symmetric!
>>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]]))
>>> k = np.array([2])
>>> precomp = gpy.fixed_dof_solve_precompute(A, k=k)
>>>
>>> b = np.array([0.,0.,0.])
>>> y = np.array([5.])
>>> u = precomp.solve(b=b, y=y)
>>> u
array([-5.,  0.,  5.])
>>>
>>> # A*u matches b, except for indices where constraints apply.
>>> A*u - b
array([0., 0., 5.])
``````
Source code in `src/gpytoolbox/fixed_dof_solve.py`
 ``` 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``` ``````def __init__(self, A, k=None): """Prepare a precomputation object to efficiently solve a linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution. For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows. This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values. Parameters ---------- A : (n,n) scipy csc matrix square matrix for the linear system k : (o,) numpy int array index vector of fixed degrees of freedom Returns ------- precomputed : precomputation object that can be used to solve the problem See Also -------- min_quad_with_fixed Examples -------- ```python >>> import gpytoolbox as gpy >>> import numpy as np >>> import scipy as sp >>> >>> # This matrix is not symmetric! >>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]])) >>> k = np.array([2]) >>> precomp = gpy.fixed_dof_solve_precompute(A, k=k) >>> >>> b = np.array([0.,0.,0.]) >>> y = np.array([5.]) >>> u = precomp.solve(b=b, y=y) >>> u array([-5., 0., 5.]) >>> >>> # A*u matches b, except for indices where constraints apply. >>> A*u - b array([0., 0., 5.]) ``` """ self.n = A.shape[0] assert A.shape[1] == self.n assert self.n>0 # self.A = A.copy() if k is None: self.o = 0 self.k = None else: self.o = k.shape[0] assert k.shape == (self.o,) assert np.min(k)>=0 and np.max(k)

### `solve(b=None, y=None)`

Solve the prefactored linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution.

For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows.

This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values.

Parameters:

Name Type Description Default
`b` `(n,) or (n,p) numpy float array`

right-hand side of the linear system

`None`
`y` `(o,) or (o,p) numpy float array`

what the degrees of freedom are fixed to, `u[k] == y` or `u[k,:] == y`

`None`

Returns:

Name Type Description
`u` `(n,) or (n,p) numpy float array such that`

`A[not k,:] * u == b[not k]` or `A[not k,:] * u == b[not k,:]` and `u[k] == y` or `u[k,:] == y`.

Examples:

``````>>> import gpytoolbox as gpy
>>> import numpy as np
>>> import scipy as sp
>>>
>>> # This matrix is not symmetric!
>>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]]))
>>> k = np.array([2])
>>> precomp = gpy.fixed_dof_solve_precompute(A, k=k)
>>>
>>> b = np.array([0.,0.,0.])
>>> y = np.array([5.])
>>> u = precomp.solve(b=b, y=y)
>>> u
array([-5.,  0.,  5.])
>>>
>>> # A*u matches b, except for indices where constraints apply.
>>> A*u - b
array([0., 0., 5.])
``````
Source code in `src/gpytoolbox/fixed_dof_solve.py`
 ```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``` ``````def solve(self, b=None, y=None): """Solve the prefactored linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution. For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows. This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values. Parameters ---------- b : (n,) or (n,p) numpy float array right-hand side of the linear system y : (o,) or (o,p) numpy float array what the degrees of freedom are fixed to, `u[k] == y` or `u[k,:] == y` Returns ------- u : (n,) or (n,p) numpy float array such that `A[not k,:] * u == b[not k]` or `A[not k,:] * u == b[not k,:]` and `u[k] == y` or `u[k,:] == y`. See Also -------- min_quad_with_fixed Examples -------- ```python >>> import gpytoolbox as gpy >>> import numpy as np >>> import scipy as sp >>> >>> # This matrix is not symmetric! >>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]])) >>> k = np.array([2]) >>> precomp = gpy.fixed_dof_solve_precompute(A, k=k) >>> >>> b = np.array([0.,0.,0.]) >>> y = np.array([5.]) >>> u = precomp.solve(b=b, y=y) >>> u array([-5., 0., 5.]) >>> >>> # A*u matches b, except for indices where constraints apply. >>> A*u - b array([0., 0., 5.]) ``` """ def cp(x): if x is None or np.isscalar(x): return 0 if len(x.shape)==1: return 0 return x.shape[1] p = max([cp(b), cp(y)]) assert b is None or np.isscalar(b) or (p==0 and b.shape==(self.n,)) or (p>0 and b.shape==(self.n,p)) assert y is None or (np.isscalar(y) and self.o>0) or (p==0 and y.shape==(self.o,)) or (p>0 and y.shape==(self.o,p)) # Get everything to full dimensions if b is None: b = 0. if np.isscalar(b): if p==0: b = np.full(self.n, b) else: b = np.full((self.n,p), b) if y is None and self.o>0: y = 0. if np.isscalar(y) and self.o>0: if p==0: y = np.full(self.o, y) else: y = np.full((self.o,p), y) # Modified rhs based on known values if self.o==0: rhs = b else: b_reduced = b[self.ki] if p==0 else b[self.ki,:] rhs = b_reduced - self.A_for_extra_b*y ured = self.solver(rhs) if self.o==0: u = ured else: if p==0: u = np.empty(self.n, dtype=np.float64) u[self.ki] = ured u[self.k] = y else: u = np.empty((self.n,p), dtype=np.float64) u[self.ki,:] = ured u[self.k,:] = y return u ``````

## `fixed_dof_solve(A, b=None, k=None, y=None)`

Solves a linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution.

For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows.

This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values.

Parameters:

Name Type Description Default
`A` `(n,n) scipy csc matrix`

square matrix for the linear system

required
`b` `(n,) or (n,p) numpy float array`

right-hand side of the linear system

`None`
`k` `(o,) numpy int array`

index vector of fixed degrees of freedom

`None`
`y` `(o,) or (o,p) numpy float array`

what the degrees of freedom are fixed to, `u[k] == y` or `u[k,:] == y`

`None`

Returns:

Name Type Description
`u` `(n,) or (n,p) numpy float array such that`

`A[not k,:] * u == b[not k]` or `A[not k,:] * u == b[not k,:]` and `u[k] == y` or `u[k,:] == y`.

`min_quad_with_fixed` for solving a quadratic programming problem with linear constraints and a symmetric matrix.

Examples:

``````>>> import gpytoolbox as gpy
>>> import numpy as np
>>> import scipy as sp
>>>
>>> # This matrix is not symmetric!
>>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]]))
>>> b = np.array([0.,0.,0.])
>>> k = np.array([2])
>>> y = np.array([5.])
>>> u = gpy.fixed_dof_solve(A, b, k, y)
>>> u
array([-5.,  0.,  5.])
>>>
>>> # A*u matches b, except for indices where constraints apply.
>>> A*u - b
array([0., 0., 5.])
``````
Source code in `src/gpytoolbox/fixed_dof_solve.py`
 ``` 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``` ``````def fixed_dof_solve(A, b=None, k=None, y=None): """Solves a linear system while fixing certain degrees of freedom for which the linear system is to be ignored during solution. For the linear system `A*u = b`, the linear system will be enforced at all the rows that don't correspond to fixed degrees of freedom, and the fixed degrees of freedom will be enforced on their respective rows. This can be used to implement finite differences with Dirichlet boundary conditions by fixing the boundary degrees of freedom to the appropriate boundary values. Parameters ---------- A : (n,n) scipy csc matrix square matrix for the linear system b : (n,) or (n,p) numpy float array right-hand side of the linear system k : (o,) numpy int array index vector of fixed degrees of freedom y : (o,) or (o,p) numpy float array what the degrees of freedom are fixed to, `u[k] == y` or `u[k,:] == y` Returns ------- u : (n,) or (n,p) numpy float array such that `A[not k,:] * u == b[not k]` or `A[not k,:] * u == b[not k,:]` and `u[k] == y` or `u[k,:] == y`. See Also -------- `min_quad_with_fixed` for solving a quadratic programming problem with linear constraints and a symmetric matrix. Examples -------- ```python >>> import gpytoolbox as gpy >>> import numpy as np >>> import scipy as sp >>> >>> # This matrix is not symmetric! >>> A = sp.sparse.csc_matrix(np.array([[1.,0.,1.],[0.,1.,0.],[0.,0.,1.]])) >>> b = np.array([0.,0.,0.]) >>> k = np.array([2]) >>> y = np.array([5.]) >>> u = gpy.fixed_dof_solve(A, b, k, y) >>> u array([-5., 0., 5.]) >>> >>> # A*u matches b, except for indices where constraints apply. >>> A*u - b array([0., 0., 5.]) ``` """ return fixed_dof_solve_precompute(A, k).solve(b, y) ``````