# `taichi.linalg`#

Taichi support module for sparse matrix operations.

class taichi.linalg.LinearOperator(matvec_kernel)#
matvec(self, x, Ax)#
taichi.linalg.MatrixFreeBICGSTAB(A, b, x, tol=1e-06, maxiter=5000, quiet=True)#

Use BiCGSTAB method to solve the linear system Ax = b, where A is implicitly represented as a LinearOperator.

Parameters:
• A (LinearOperator) – The coefficient matrix A of the linear system.

• b (Field) – The right-hand side of the linear system.

• x (Field) – The initial guess for the solution.

• maxiter (int) – Maximum number of iterations.

• atol – Tolerance(absolute) for convergence.

• quiet (bool) – Switch to turn on/off iteration log.

taichi.linalg.MatrixFreeCG(A, b, x, tol=1e-06, maxiter=5000, quiet=True)#

Use conjugate-gradient method to solve the linear system Ax = b, where A is implicitly represented as a LinearOperator.

Parameters:
• A (LinearOperator) – The coefficient matrix A of the linear system.

• b (Field) – The right-hand side of the linear system.

• x (Field) – The initial guess for the solution.

• maxiter (int) – Maximum number of iterations.

• atol – Tolerance(absolute) for convergence.

• quiet (bool) – Switch to turn on/off iteration log.

class taichi.linalg.SparseCG(A, b, x0=None, max_iter=50, atol=1e-06)#

Use conjugate-gradient method to solve the linear system Ax = b, where A is SparseMatrix.

Parameters:
• A (SparseMatrix) – The coefficient matrix A of the linear system.

• b (numpy ndarray, taichi Ndarray) – The right-hand side of the linear system.

• x0 (numpy ndarray, taichi Ndarray) – The initial guess for the solution.

• max_iter (int) – Maximum number of iterations.

• atol – Tolerance(absolute) for convergence.

solve(self)#
class taichi.linalg.SparseMatrix(n=None, m=None, sm=None, dtype=f32, storage_format='col_major')#

Taichi’s Sparse Matrix class

A sparse matrix allows the programmer to solve a large linear system.

Parameters:
• n (int) – the first dimension of a sparse matrix.

• m (int) – the second dimension of a sparse matrix.

• sm (SparseMatrix) – another sparse matrix that will be built from.

build_from_ndarray(self, ndarray)#

Build the sparse matrix from a ndarray.

Parameters:

ndarray (Union[ti.ndarray, ti.Vector.ndarray, ti.Matrix.ndarray]) – the ndarray to build the sparse matrix from.

Raises:

TaichiRuntimeError – If the input is not a ndarray or the length is not divisible by 3.

Example::
```>>> N = 5
>>> triplets = ti.Vector.ndarray(n=3, dtype=ti.f32, shape=10, layout=ti.Layout.AOS)
>>> @ti.kernel
>>> def fill(triplets: ti.types.ndarray()):
>>>     for i in range(N):
>>>        triplets[i] = ti.Vector([i, (i + 1) % N, i+1], dt=ti.f32)
>>> fill(triplets)
>>> A = ti.linalg.SparseMatrix(n=N, m=N, dtype=ti.f32)
>>> A.build_from_ndarray(triplets)
>>> print(A)
[0, 1, 0, 0, 0]
[0, 0, 2, 0, 0]
[0, 0, 0, 3, 0]
[0, 0, 0, 0, 4]
[5, 0, 0, 0, 0]
```
mmwrite(self, filename)#

Writes the sparse matrix to Matrix Market file-like target.

Parameters:

filename (str) – the file name to write the sparse matrix to.

transpose(self)#

Sparse Matrix transpose.

Returns:

The transposed sparse mastrix.

class taichi.linalg.SparseMatrixBuilder(num_rows=None, num_cols=None, max_num_triplets=0, dtype=f32, storage_format='col_major')#

A python wrap around sparse matrix builder.

Use this builder to fill the sparse matrix.

Parameters:
• num_rows (int) – the first dimension of a sparse matrix.

• num_cols (int) – the second dimension of a sparse matrix.

• max_num_triplets (int) – the maximum number of triplets.

• dtype (ti.dtype) – the data type of the sparse matrix.

• storage_format (str) – the storage format of the sparse matrix.

build(self, dtype=f32, _format='CSR')#

Create a sparse matrix using the triplets

print_triplets(self)#

Print the triplets stored in the builder

class taichi.linalg.SparseSolver(dtype=f32, solver_type='LLT', ordering='AMD')#

Sparse linear system solver

Use this class to solve linear systems represented by sparse matrices.

Parameters:
• solver_type (str) – The factorization type.

• ordering (str) – The method for matrices re-ordering.

analyze_pattern(self, sparse_matrix)#

Reorder the nonzero elements of the matrix, such that the factorization step creates less fill-in.

Parameters:

sparse_matrix (SparseMatrix) – The sparse matrix to be analyzed.

compute(self, sparse_matrix)#

This method is equivalent to calling both analyze_pattern and then factorize.

Parameters:

sparse_matrix (SparseMatrix) – The sparse matrix to be computed.

factorize(self, sparse_matrix)#

Do the factorization step

Parameters:

sparse_matrix (SparseMatrix) – The sparse matrix to be factorized.

info(self)#

Check if the linear systems are solved successfully.

Returns:

True if the solving process succeeded, False otherwise.

Return type:

bool

solve(self, b)#

Computes the solution of the linear systems. :param b: The right-hand side of the linear systems. :type b: numpy.array or Field

Returns:

The solution of linear systems.

Return type:

numpy.array

exception taichi.linalg.TaichiRuntimeError#

Bases: `RuntimeError`

Thrown when the compiled program cannot be executed due to unspecified reasons.

class args#
with_traceback()#

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

exception taichi.linalg.TaichiTypeError#

Bases: `TaichiCompilationError`, `TypeError`

Thrown when a type mismatch is found during compilation.

class args#
with_traceback()#

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.