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 Incoming links: Stable construction of a Givens rotation

Let

r = x_{i}^{2} + x_{j}^{2}

. If

r > 0

, choose

θ

so that

cos θ = \frac{x _{i}}{r}, sin θ = \frac{x _{j}}{r} .

Such an angle exists because the two ratios have squared sum one. Then the changed coordinates are

y_{i} = \frac{x _{i}^{2} + x _{j}^{2}}{r} = r, y_{j} = - \frac{x _{j} x _{i}}{r} + \frac{x _{i} x _{j}}{r} = 0.

All other coordinates satisfy

y_{k} = x_{k}

. If

r = 0

, both input coordinates are zero; take

θ = 0

, giving the identity matrix on this pair. In all cases,

y_{i} = x_{i}^{2} + x_{j}^{2}, y_{j} = 0, y_{k} = x_{k} (k \neq = i, j) .

This fixes a nonnegative retained coordinate and handles division by zero explicitly. For floating-point computation, the stable construction of a Givens rotation uses scaled Euclidean norm evaluation to avoid unnecessarily overflowing or underflowing the squares.