vectorNorm
Syntax
vectorNorm(x, [ord], [axis], [keepDims])
Arguments
x is a vector or matrix of any numeric type except DECIMAL. It cannot be empty.
ord (optional) is an INT, STRING or floating-point scalar indicating the order of norm. Note:
- When ord is a string, it must be: 'inf', '-inf', 'nuc', or 'fro'.
- When ord is less than 1, the result is technically not a mathematical "norm", but it may still be useful for various numerical purposes.
The following describes the method for computing the norm based on different x and ord:
| ord | norm for vectors | norm for matrices |
|---|---|---|
| None | 2-norm | Frobenius norm |
| 0 | sum(x != 0) | - |
| -1 | sum(abs(x)^ord)^(1/ord) | min(sum(abs(x), axis=0)) |
| 1 | sum(abs(x)^ord)^(1/ord) | max(sum(abs(x), axis=0)) |
| -2 | sum(abs(x)^ord)^(1/ord) | 2-norm (largest sing. value) |
| 2 | sum(abs(x)^ord)^(1/ord) | smallest singular value |
| inf | max(abs(x)) | max(sum(abs(x), axis=1)) |
| -inf | min(abs(x)) | min(sum(abs(x), axis=1)) |
| nuc | - | nuclear norm |
| fro | - | Frobenius norm |
| other | sum(abs(x)^ord)^(1/ord) | - |
axis (optional) is an integer vector or scalar indicating the direction along which to compute the norm. It cannot contain empty elements.
- When x is a vector, axis can only be 0.
- When x is a matrix, axis:
- Has a length of no more than 2.
- Cannot contain duplicate elements.
- Has elements with values of 0 or 1.
keepDims (optional): A boolean scalar indicating whether the returned result should maintain the same form as x. The default is false.
Details
The function computes a matrix or vector norm. Note that it’s not recommended to use
vectorNorm in SQL statements.
Return Value: An INT, LONG, or DOUBLE scalar, vector, or matrix.
Examples
When x is a vector, compute norms:
x = 1..4
vectorNorm(x) // Output: 5.477225575051661
vectorNorm(x, keepDims=true) // Output: 5.477225575051661
vectorNorm(x, ord=1) // Output: 10
vectorNorm(x, ord=1, axis=0) // Output: 10
vectorNorm(x, ord=-1) // Output 0.4800000000000001
vectorNorm(x, ord=-1, axis=0) // Output double: 10
vectorNorm(x, ord="inf", axis=0) //Output: 4
vectorNorm(x, ord="-inf", axis=0) //Output: 1
vectorNorm(x, ord=3) // Output: 4.641588833612778
vectorNorm(x, ord=-20.689) // Output: 0.9999999714010688
vectorNorm(x, ord="fro") // throw exception
vectorNorm(x, ord="nuc") // throw exceptionWhen x is a matrix, compute norms:
x = 1..4$2:2
vectorNorm(x) // Output: 5.477225575051661
vectorNorm(x, keepDims=true) // Output: 5.477225575051661
vectorNorm(x, ord=1) // Output: 7
vectorNorm(x, ord=1, axis=0) // Output: 3 7
vectorNorm(x, ord=1, axis=1) // Output: 4 6
vectorNorm(x, ord=-1, axis=0) // Output: 0.6666666666666666 1.7142857142857144
vectorNorm(x, ord=-1, axis=1) // Output: 0.75 1.3333333333333333
vectorNorm(x, ord=1, axis=(0 1)) // Output: 7
vectorNorm(x, ord=1, axis=(1 0)) // Output: 6
vectorNorm(x, ord=-1, axis=(0 1)) // Output: 3
vectorNorm(x, ord=-1, axis=(1 0)) // Output: 4
vectorNorm(x, ord="inf", axis=(0 1)) // Output: 4
vectorNorm(x, ord="inf", axis=(1 0)) // Output: 3
vectorNorm(x, ord="-inf", axis=(0 1)) // Output: 6
vectorNorm(x, ord="-inf", axis=(1 0)) // Output: 7
vectorNorm(x, ord="fro", axis=(1 0)) // Output: 5.477225575051661
vectorNorm(x, ord="fro", axis=(0 1)) // Output: 5.477225575051661
vectorNorm(x, ord=-2, axis=(1 0)) // Output: 0.3659661906262574
vectorNorm(x, ord=-2, axis=(0 1)) // Output: 0.3659661906262574
vectorNorm(x, ord=2, axis=(1 0)) // Output: 5.464985704219043
vectorNorm(x, ord=2, axis=(0 1)) // Output: 5.464985704219043
vectorNorm(x, ord="nuc", axis=(1 0)) // Output: 5.8309518948453
vectorNorm(x, ord="nuc", axis=(0 1)) // Output: 5.8309518948453
vectorNorm(x, ord=3) // throw exception