Before we can understand why the conjugate-gradient method is so fast, we need to see why the steepest-descent method is so slow. The process of selecting is called line search," but for a linear problem like the one we have chosen here, we hardly recognize choosing as searching a line. A more graphic understanding of the whole process is possible in a two-dimensional space where the vector of unknowns x has just two components, x1 and x2. Then the size of the residual vector can be displayed with a contour plot in the plane of (x1,x2). Visualize a contour map of a mountainous terrain. The gradient is perpendicular to the contours. Contours and gradients are curved lines. In the steepest-descent method we start at a point and compute the gradient direction at that point. Then we begin a straight-line descent in that direction. The gradient direction curves away from our direction of travel, but we continue on our straight line until we have stopped descending and are about to ascend. There we stop, compute another gradient vector, turn in that direction, and descend along a new straight line. The process repeats until we get to the bottom, or until we get tired.