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MathidiaMathematics Academy
Numerical analysis
Lesson 43 of 75
Standard practice

Numerical least squares

Objective. Use QR methods for stable fitting.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Numerical least squares finds the vector x minimising ‖Ax − b‖² for an overdetermined system, using a QR factorisation of A for numerical stability rather than forming the ill-conditioned normal equations AᵀAx = Aᵀb directly. Writing A = QR with Q orthogonal and R upper triangular, the problem reduces to solving Rx = Qᵀb by back substitution, since ‖Ax − b‖ = ‖Rx − Qᵀb‖. For example, fitting a line to points (0,1), (1,2), (2,2) sets up a 3×2 matrix A whose QR factorisation yields a stable triangular system for the slope and intercept. QR avoids squaring the condition number that normal equations introduce.

Worked example

Using least squares, fit y = c (a constant) to the data points (1,2), (2,4), (3,3), finding the value of c that minimises the sum of squared residuals.

Diagram of a square
  1. 1Model: A is a column of ones, so the normal equation becomes 3c = Σyᵢ.
  2. 2Sum the y-values: 2 + 4 + 3 = 9.
  3. 3Solve 3c = 9 to give c = 3.
  4. 4Check: residuals are −1, 1, 0, summing to zero as required for a least-squares constant fit.

Answer: c = 3

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Common mistake. Students solve AᵀAx = Aᵀb directly for a fit and get inaccurate coefficients when A is ill-conditioned, because squaring A in AᵀA doubles the condition number; using QR decomposition avoids this amplification of rounding error.

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