Is QR decomposition a direct linear system solving method? If I have $AX=b$ and I wand to solve this system for $X$ via QR decomposition. Is the $QR$ decomposition regarded as as part of the family of direct solve methods?
It seems to me that the solve in itself (when $Q$ and $R$ are known) is direct via back-substitution. But it also seems that obtaining $Q$ and $R$ can be iterative (depending on the method used).
What's the official status of $QR$ decomposition?
 A: Brief answers:
1) Solving a linear system $Ax = b$ using the $QR$ decomposition of the matrix $A$ is a direct method.
2) To my knowledge, all algorithms for computing the QR decomposition of a dense $n$ by $n$ matrix are classified as direct methods.
Detailed explanation with examples:
A method is function $M : \Omega \rightarrow V$ which maps the input $\omega \in \Omega$ to the result $v \in V$.
A method is said to be direct if it returns the result after a number of arithmetic operations which is independent of the input.
Example: The solution of the $n$ by $n$ real, upper triangular linear system $Ux=b$ using backward substitution. Here $\Omega = \mathbb R^{n \times n} \times \mathbb R^n$ and $V = \mathbb{R}^n$. The input $(U,b)$ is mapped to the solution $x$. The cost is $n^2$ arithmetic operations.
Newton's method for solving a nonlinear equation $$f(x) = \alpha$$ is an iterative method. However, there are problems for which an initial guess can be constructed, such that the method converges to machine precision after a finite number of steps, which is independent of $\alpha$.
Example: The solution of the equation $x^2 = \alpha > 0$. The general case of $\alpha > 0$ can be reduced to the special case of $\alpha \in [1,4]$ by exploiting the basic properties of floating point numbers. A good initial guess of the form $x_0(s) = a s + b$ can be constructed for the special case using the best uniform approximation of $s \rightarrow \sqrt{s}$ on the interval $[1,4]$.
Of course, the majority of iterative methods cannot be classified as direct methods.
Example: The solution of an $n$ by $n$ linear system $Ax=b$ where $A$ is a sparse symmetric positive definite matrix using the conjugate gradient algorithm without a preconditioner. 
The QR decomposition of a matrix can be computed using Gram-Schmidt with reorthogonalization. In theory, one reorthogonalizes until convergence, but in practice only two such steps are required. Hence this method is (correctly) regarded as direct method.
The QR decomposition can also be computed using Givens rotations or Householder reflectors. These methods are direct methods.
