Numerical Methods · Floating-point arithmetic, conditioning, and stability
A decimal floating-point system uses base β = 10, precision t = 4, and exponent range…
Problem
A decimal floating-point system uses base \(\beta = 10\), precision \(t = 4\), and exponent range \(-2 \le e \le 3\). Normalized significands satisfy \(1 \le d_0.d_1 d_2 d_3 < 10\). List every positive machine number of the form \(m \times 10^{e}\) with \(e = 0\). Then compute the absolute and relative gaps between consecutive machine numbers at \(x = 1\) and at \(x = 9.999 \times 10^{0}\).
Hint
A normalized four-digit decimal significand is a number \(d_0.d_1 d_2 d_3\) with leading digit nonzero.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
Create a free account to check your answer and see the solution. Create a free account.
More Numerical Methods practice problems
- In four-digit decimal floating-point arithmetic with rounding to nearest, compute…Floating-point arithmetic, conditioning, and stability
- Let g be continuously differentiable on an open interval containing a fixed point…Root finding and nonlinear systems
- Let A∈R^(n× n) be strictly row-diagonally dominant, and let G_J=D^(-1)(L+U) be the…Linear systems, least squares, and eigenproblems
- The values f(1.0)=2.7183, f(1.1)=3.0042, f(1.2)=3.3201, and f(1.3)=3.6693 are given…Interpolation and approximation
- A fixed-point rearrangement is useless unless it contracts on an invariant intervalRoot finding and nonlinear systems
- Two conjugate-gradient steps close an SPD two-by-two residualLinear systems, least squares, and eigenproblems
- Matching values and derivatives needs repeated-node divided differencesInterpolation and approximation
- Two Gauss nodes integrate cubics exactly and miss a concrete quarticNumerical differentiation and quadrature
- Linear shooting hits the far boundary in one sensitivity stepNumerical ODE and introductory PDE methods