SOLVE_BANDED

Solves a linear system where the coefficient matrix A is banded. Banded matrices only have non-zero elements in a few diagonals around the main diagonal, which allows for significantly reduced storage and computation time. The matrix must be provided in the ‘ab’ format used by LAPACK.

Excel Usage

=SOLVE_BANDED(l_diags, u_diags, ab_matrix, b_vector)

l_diags (int, required): Number of non-zero lower diagonals.
u_diags (int, required): Number of non-zero upper diagonals.
ab_matrix (list[list], required): The banded matrix in ‘ab’ format (shape is l + u + 1 by M).
b_vector (list[list], required): 2D array representing the right-hand side.

Returns (list[list]): 2D array representing the solution x.

Example 1: Solve diagonal matrix (special case of banded)

Inputs:

l_diags	u_diags	ab_matrix		b_vector
0	0	2	3	4
				9

Excel formula:

=SOLVE_BANDED(0, 0, {2,3}, {4;9})

Expected output:

Result
2
3

Example 2: Solve tridiagonal system with identity-like right-hand side

Inputs:

l_diags	u_diags	ab_matrix			b_vector
1	1	0	-1	-1	1
		2	2	2	0
		-1	-1	0	1

Excel formula:

=SOLVE_BANDED(1, 1, {0,-1,-1;2,2,2;-1,-1,0}, {1;0;1})

Expected output:

Result
1
1
1

Example 3: Solve banded system (1 lower, 1 upper)

Inputs:

l_diags	u_diags	ab_matrix			b_vector
1	1	0	1	1	1
		2	2	2	1
		1	1	0	1

Excel formula:

=SOLVE_BANDED(1, 1, {0,1,1;2,2,2;1,1,0}, {1;1;1})

Expected output:

Result
0.5
0
0.5

Example 4: Solve upper-banded diagonal case with two right-hand sides

Inputs:

l_diags	u_diags	ab_matrix		b_vector
0	0	4	5	8	4
				10	15

Excel formula:

=SOLVE_BANDED(0, 0, {4,5}, {8,4;10,15})

Expected output:

Result
2	1
2	3

Python Code

import numpy as np
from scipy.linalg import solve_banded as scipy_solve_banded

def solve_banded(l_diags, u_diags, ab_matrix, b_vector):
    """
    Solve the equation Ax = b for x, assuming A is a banded matrix.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_banded.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        l_diags (int): Number of non-zero lower diagonals.
        u_diags (int): Number of non-zero upper diagonals.
        ab_matrix (list[list]): The banded matrix in 'ab' format (shape is l + u + 1 by M).
        b_vector (list[list]): 2D array representing the right-hand side.

    Returns:
        list[list]: 2D array representing the solution x.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        ab_matrix = to2d(ab_matrix)
        b_vector = to2d(b_vector)

        try:
            ab = np.array(ab_matrix, dtype=float)
            b = np.array(b_vector, dtype=float)
        except (ValueError, TypeError):
            return "Error: matrix entries must contain numeric values"

        try:
            x = scipy_solve_banded((int(l_diags), int(u_diags)), ab, b)
        except Exception as e:
            return f"Error: {str(e)}"

        return x.tolist()

    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

l_diags *

Number of non-zero lower diagonals.

u_diags *

Number of non-zero upper diagonals.

ab_matrix *

The banded matrix in 'ab' format (shape is l + u + 1 by M).

b_vector *

2D array representing the right-hand side.