TriangularSolve

Solves systems of linear equations with upper or lower triangular matrices by backsubstitution.

`matrix` is a tensor of shape `[..., M, M]` whose inner-most 2 dimensions form square matrices. If `lower` is `True` then the strictly upper triangular part of each inner-most matrix is assumed to be zero and not accessed. If `lower` is False then the strictly lower triangular part of each inner-most matrix is assumed to be zero and not accessed. `rhs` is a tensor of shape `[..., M, N]`.

The output is a tensor of shape `[..., M, N]`. If `adjoint` is `True` then the innermost matrices in `output` satisfy matrix equations `matrix[..., :, :] * output[..., :, :] = rhs[..., :, :]`. If `adjoint` is `False` then the strictly then the innermost matrices in `output` satisfy matrix equations `adjoint(matrix[..., i, k]) * output[..., k, j] = rhs[..., i, j]`.

Note, the batch shapes for the inputs only need to broadcast.

Example:

a = tf.constant([[3,  0,  0,  0],
                  [2,  1,  0,  0],
                  [1,  0,  1,  0],
                  [1,  1,  1,  1]], dtype=tf.float32)
 
 b = tf.constant([[4],
                  [2],
                  [4],
                  [2]], dtype=tf.float32)
 
 x = tf.linalg.triangular_solve(a, b, lower=True)
 x
 # <tf.Tensor: shape=(4, 1), dtype=float32, numpy=
 # array([[ 1.3333334 ],
 #        [-0.66666675],
 #        [ 2.6666665 ],
 #        [-1.3333331 ]], dtype=float32)>
 
 # in python3 one can use `a@x`
 tf.matmul(a, x)
 # <tf.Tensor: shape=(4, 1), dtype=float32, numpy=
 # array([[4.       ],
 #        [2.       ],
 #        [4.       ],
 #        [1.9999999]], dtype=float32)>