Newton's method fails when the function is flat because the local slope (derivative) is zero or near-zero. Since the method relies on drawing a tangent line and finding its x-intercept, a horizontal tangent does not intersect the x-axis at a useful new point, preventing progress toward the root.
Conditions: The function is locally flat at the current point.; The slope is too small or zero to provide a useful Newton step.
Newton's method fails when the function is flat because the local slope (derivative) is zero or near-zero. Since the method relies on drawing a tangent line and finding its x-intercept, a horizontal tangent does not intersect the x-axis at a useful new point, preventing progress toward the root.
Conditions: The function is locally flat at the current point.; The slope is too small or zero to provide a useful Newton step.
Newton's method uses an iterative geometric procedure. Starting from an initial guess, it moves vertically to the function graph, computes the local slope, draws a tangent line, and takes that tangent's x-intercept as the next guess.
Conditions: An initial guess x must be chosen.; The function has a usable slope at the current point.; The tangent line must intersect the horizontal zero line to produce the next guess.
Newton's method uses an iterative geometric procedure. Starting from an initial guess, it moves vertically to the function graph, computes the local slope, draws a tangent line, and takes that tangent's x-intercept as the next guess.
Conditions: An initial guess x must be chosen.; The function has a usable slope at the current point.; The tangent line must intersect the horizontal zero line to produce the next guess.