Introduction to numerics

The numerics package provides NumPy-like numerical computing for Maxima. It wraps BLAS and LAPACK via the magicl library, giving Maxima users access to fast dense linear algebra, element-wise array operations, and statistical aggregations.

Requirements: SBCL (Steel Bank Common Lisp). Other Lisp implementations are not supported.

Key concepts

ndarray is the central data type – an opaque handle wrapping a dense numeric array. Unlike Maxima matrices, ndarrays:

Use 0-based indexing (like NumPy, C, and most programming languages)
Store elements as double-float (64-bit IEEE 754)
Use column-major memory layout for BLAS compatibility
Are mutable – functions like np_set modify the array in place

Quick start

(%i1) load("numerics")$
(%i2) A : ndarray(matrix([1, 2], [3, 4]));
(%o2)            ndarray([2, 2], DOUBLE-FLOAT)
(%i3) B : np_inv(A);
(%o3)            ndarray([2, 2], DOUBLE-FLOAT)
(%i4) np_to_matrix(np_matmul(A, B));
(%o4)            matrix([1.0, 0.0], [0.0, 1.0])

Creating arrays

(%i1) np_zeros([3, 3]);
(%o1)            ndarray([3, 3], DOUBLE-FLOAT)
(%i2) np_ones([2, 4]);
(%o2)            ndarray([2, 4], DOUBLE-FLOAT)
(%i3) np_arange(5);
(%o3)            ndarray([5], DOUBLE-FLOAT)
(%i4) np_linspace(0, 1, 5);
(%o4)            ndarray([5], DOUBLE-FLOAT)

Element access

(%i1) A : ndarray(matrix([10, 20, 30], [40, 50, 60]));
(%o1)            ndarray([2, 3], DOUBLE-FLOAT)
(%i2) np_ref(A, 0, 2);
(%o2)                          30.0
(%i3) np_set(A, 1, 1, 99.0);
(%o3)            ndarray([2, 3], DOUBLE-FLOAT)
(%i4) np_ref(A, 1, 1);
(%o4)                          99.0

Element-wise operations

All element-wise functions support ndarray-ndarray and ndarray-scalar combinations:

(%i1) A : ndarray(matrix([1, 2], [3, 4]));
(%o1)            ndarray([2, 2], DOUBLE-FLOAT)
(%i2) np_add(A, 10);
(%o2)            ndarray([2, 2], DOUBLE-FLOAT)
(%i3) np_mul(A, A);
(%o3)            ndarray([2, 2], DOUBLE-FLOAT)
(%i4) np_sqrt(A);
(%o4)            ndarray([2, 2], DOUBLE-FLOAT)

Linear algebra

(%i1) A : ndarray(matrix([2, 1], [5, 3]));
(%o1)            ndarray([2, 2], DOUBLE-FLOAT)
(%i2) np_det(A);
(%o2)                          1.0
(%i3) [U, S, Vt] : np_svd(A);
(%o3)        [ndarray, ndarray, ndarray]
(%i4) [Q, R] : np_qr(A);
(%o4)              [ndarray, ndarray]

Converting back to Maxima

(%i1) A : np_linspace(0, 1, 5);
(%o1)            ndarray([5], DOUBLE-FLOAT)
(%i2) np_to_list(A);
(%o2)           [0.0, 0.25, 0.5, 0.75, 1.0]
(%i3) M : ndarray(matrix([1, 2], [3, 4]));
(%o3)            ndarray([2, 2], DOUBLE-FLOAT)
(%i4) np_to_matrix(M);
(%o4)         matrix([1.0, 2.0], [3.0, 4.0])

Definitions for numerics

Conversion

`ndarray` (a) / ndarray (a, complex) — Function

Convert a Maxima matrix or list to an ndarray.

If a is a Maxima matrix, it is converted element-by-element into a double-float ndarray with column-major layout. If a is a Maxima list, it creates a 1D ndarray unless a shape argument is provided. Pass complex as the last argument to create a complex-double-float array.

If a is already an ndarray, it is returned unchanged.

Calling forms:

ndarray(matrix) – convert a Maxima matrix to a 2D ndarray
ndarray(matrix, complex) – convert to a complex 2D ndarray
ndarray(list) – convert a flat Maxima list to a 1D ndarray
ndarray(list, shape) – convert a list and reshape to the given dimensions
ndarray(list, shape, complex) – convert a list to a complex ndarray with shape

The shape argument is a Maxima list of dimensions, e.g. [2, 3]. The total number of elements in the list must match the product of the dimensions.

Elements are filled in row-major order from the list, matching the convention that ndarray([1,2,3,4], [2,2]) produces a matrix with first row [1, 2] and second row [3, 4].

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