feat: add lapack/base/dlartg

[aayush0325]

Description

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This pull request:

• adds lapack/base/dlartg

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• Read, understood, and followed the contributing guidelines.

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[stdlib-bot]

Coverage Report

Package	Statements	Branches	Functions	Lines
lapack/base/dlartg	$\color{green}355/355$ $\color{green}+100.00\%$	$\color{green}17/17$ $\color{green}+100.00\%$	$\color{green}3/3$ $\color{green}+100.00\%$	$\color{green}355/355$ $\color{green}+100.00\%$

The above coverage report was generated for the changes in this PR.

[kgryte]

Suggested change

	`dlartg` constructs a Givens rotation for a two-dimensional vector such that the following equation is satisfied:
	This routine constructs a Givens rotation for a two-dimensional vector such that the following equation is satisfied:

[kgryte]

@aayush0325 I'm a little confused. What is meant by a "two-dimensional" vector? Vectors are, by definition, one dimensional. Are you meaning a column matrix?

[kgryte]

Or maybe you are intending a two-element vector?

[kgryte]

Suggested change

	- **F**: single element array representing the first component of the vector to be rotated as a [`Float64Array`][mdn-float64array].
	- **G**: single element array representing the second component of the vector to be rotated as a [`Float64Array`][mdn-float64array].
	- **C**: single element array overwritten by the cosine of the rotation as a [`Float64Array`][mdn-float64array].
	- **S**: single element array overwritten by the sine of the rotation as a [`Float64Array`][mdn-float64array].
	- **R**: single element array overwritten by the length of the rotated vector as a [`Float64Array`][mdn-float64array].
	- **F**: [`Float64Array`][mdn-float64array] containing a single element representing the first component of the vector to be rotated.
	- **G**: [`Float64Array`][mdn-float64array] containing a single element representing the second component of the vector to be rotated
	- **C**: [`Float64Array`][mdn-float64array] containing a single element which is overwritten by the cosine of the rotation.
	- **S**: [`Float64Array`][mdn-float64array] containing a single element which is overwritten by the sine of the rotation.
	- **R**: [`Float64Array`][mdn-float64array] containing a single element which is overwritten by the length of the rotated vector.

[kgryte]

Same comments as above.

[kgryte]

Suggested change

	- **offsetF**: starting index for `F`.
	- **offsetF**: index of the element in `F`.

It is somewhat awkward to say "starting index" when we are talking about only a single element. We can be more precise, IMO.

Applies here and below and likely elsewhere throughout this PR.

[kgryte]

For the descriptions, all the same comments.

[kgryte]

Prefer the alias copysign. sign means something else in other contexts.

[kgryte]

Prefer two calls to fast max, rather than using maxn; e.g., max( max( safmin, f1 ), g1 ). Variadic functions are likely to end up in de-optimization hell. Here, we want to encourage compiler inlining.

[kgryte]

This looks like hypot. Prefer that to avoid unnecessary underflow/overflow. I am surprised that LAPACK does not use hypot and/or the LAPACK equivalent routine.

[kgryte]

Same thing. This is hypot.