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npplus.solveper module

Periodic variants of solve_banded.

Adds corner elements to banded matrix connecting final elements of x to first equations and first elements of x to final equations.

Also include solves_banded, a variant of solveh_banded that does not require a positive definite symmetric matrix.

solve_periodic(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False, check_finite=True)[source]

Variant of solve_banded which includes matrix corner elements.

Layout of banded matrix ab is:

ab[u+i-j, j] == a[i, j]
ab[i, j] == a[i+j-u, j]

Example (l,u)=(2,1) for 6x6:

A50  a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55
a10  a21  a32  a43  a54  A05
a20  a31  a42  a53  A04  A15

A50   0    0    x    x    x    copy last row before first
a00  a01   0    0   A04  A05  <-- matrix begins here
a10  a11  a12   0    0   A15
a20  a21  a22  a23   0    0
 0   a31  a32  a33  a34   0
 0    0   a42  a43  a44  a45
A50   0    0   a53  a54  a55  <-- matrix ends here
 x    x    0    0   A04  A05   copy first two rows after last
 x    x    x    0    0   A15

See also

scipy.linalg.solve_banded

for description of parameters and returns

Notes

Outline of the algorithm:

  1. Solve ab without the corner elements using solve_banded.

  2. Solve ab without corner elements for b=0 except 1 for each of the first u equations and last l equations, a total of u+l solves with one more call to solve_banded.

  3. Construct a dense u+l square matrix and solve to find the linear combination of the results of (2) which compensates for the corner elements omitted in (1).

solves_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False, check_finite=True)[source]

Variant of solve_banded for arbitrary symmetric matrices.

Upper form unless keyword lower is false. Upper and lower forms are the same as scipy.linalg.solveh_banded, except input to solves_periodic need not be positive definite, and the entire ab array is used:

A40  A51  a02  a13  a24  a35   upper form
A50  a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55

a00  a11  a22  a33  a44  a55   lower form
a10  a21  a32  a43  a54  A05
a20  a31  a42  a53  A04  A15

See also

scipy.linalg.solveh_banded

for description of parameters and returns

Notes

Use solveh_banded if the matrix is known to be positive definite.

solves_periodic(ab, b, overwrite_ab=False, overwrite_b=False, lower=False, check_finite=True)[source]

Variant of solve_periodic for symmetric matrices.

Upper form unless keyword lower is false. Upper and lower forms are the same as scipy.linalg.solveh_banded, except input to solves_periodic need not be positive definite, and the entire ab array is used:

A40  A51  a02  a13  a24  a35   upper form
A50  a01  a12  a23  a34  a45
a00  a11  a22  a33  a44  a55

a00  a11  a22  a33  a44  a55   lower form
a10  a21  a32  a43  a54  A05
a20  a31  a42  a53  A04  A15

See also

scipy.linalg.solveh_banded

for description of parameters and returns

solve_periodic

for more details