multiply

multiply(x1: array, x2: array, /) → array

Calculates the product for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Floating-point multiplication is not always associative due to finite precision.

Parameters:

  • x1 (array) – first input array. Should have a numeric data type.

  • x2 (array) – second input array. Must be compatible with x1 (see Broadcasting). Should have a numeric data type.

Returns:

out (array) – an array containing the element-wise products. The returned array must have a data type determined by Type Promotion Rules.

Notes

Special cases

For real-valued floating-point operands,

  • If either x1_i or x2_i is NaN, the result is NaN.

  • If x1_i is either +infinity or -infinity and x2_i is either +0 or -0, the result is NaN.

  • If x1_i is either +0 or -0 and x2_i is either +infinity or -infinity, the result is NaN.

  • If x1_i and x2_i have the same mathematical sign, the result has a positive mathematical sign, unless the result is NaN. If the result is NaN, the “sign” of NaN is implementation-defined.

  • If x1_i and x2_i have different mathematical signs, the result has a negative mathematical sign, unless the result is NaN. If the result is NaN, the “sign” of NaN is implementation-defined.

  • If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is a signed infinity with the mathematical sign determined by the rule already stated above.

  • If x1_i is either +infinity or -infinity and x2_i is a nonzero finite number, the result is a signed infinity with the mathematical sign determined by the rule already stated above.

  • If x1_i is a nonzero finite number and x2_i is either +infinity or -infinity, the result is a signed infinity with the mathematical sign determined by the rule already stated above.

  • In the remaining cases, where neither infinity nor NaN is involved, the product must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the result is a zero of appropriate mathematical sign.

For complex floating-point operands, multiplication is defined according to the following table. For real components a and c and imaginary components b and d,

c

dj

c + dj

a

a * c

(a*d)j

(a*c) + (a*d)j

bj

(b*c)j

-(b*d)

-(b*d) + (b*c)j

a + bj

(a*c) + (b*c)j

-(b*d) + (a*d)j

special rules

In general, for complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table.

When a, b, c, or d are all finite numbers (i.e., a value other than NaN, +infinity, or -infinity), multiplication of complex floating-point operands should be computed as if calculated according to the textbook formula for complex number multiplication

\[(a + bj) \cdot (c + dj) = (ac - bd) + (bc + ad)j\]

When at least one of a, b, c, or d is NaN, +infinity, or -infinity,

  • If a, b, c, and d are all NaN, the result is NaN + NaN j.

  • In the remaining cases, the result is implementation dependent.

Note

For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is NaN, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex multiplication. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex multiplication according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations.

Changed in version 2022.12: Added complex data type support.