# 21. Underdetermined Systems

## Abstract

*I'm thinking of two numbers. Their average is 3. What are the numbers?*

*The World's Simplest Impossible Problem*(1990)

*This problem arises in important algorithms used in mathematical programming … In these cases, B is usually very large and sparse and, because of storage difficulties, it is often uneconomical to store and access Q*

_{1}…

*Sometimes it has been thought that*[

*the seminormal equations method*]

*could be disastrously worse than*[

*the Q method*] …

*It is the purpose of this note to show that such algorithms are numerically quite satisfactory*.

*An Error Analysis of a Method for Solving Matrix Equations*(1973)

**21.1. Solution Methods**

*Ax*=

*b*, where

*A*∈ ℝ

^{m × n}with

*m*≤

*n*. The system can be analysed using a QR factorization

*Q*∈ ℝ

^{n × n}is orthogonal and

*R*∈ ℝ

^{m × m}is upper triangular. (We could, alternatively, use an LQ factorization of

*A*, but we will keep to the standard notation.) We have

*A*has full rank then

*y*

_{1}=

*R*

^{−T}

*b*is uniquely determined and all solutions of

*Ax*=

*b*are given by

*x*

_{LS}that minimizes ‖

*x*‖

_{2}is obtained by setting

*y*

_{2}= 0. We have

*A*

^{+}=

*A*

^{T}(

*AA*

^{T})

^{−1}is the pseudo-inverse of

*A*. Hence

*x*

_{LS}can be characterized as

*x*

_{ls}=

*A*

^{T}

*y*, where

*y*solves the

*normal equations*

*AA*

^{T}

*y*=

*b*.

*x*

_{LS}. We will refer to this method as the “Q method”. When

*A*is large and sparse it is desirable to avoid storing and accessing

*Q*, which can be expensive. An alternative method with this property uses the QR factorization (21.1) but computes

*x*

_{LS}as

*x*

_{LS}=

*A*

^{T}

*y*, where

*seminormal equations*(SNE). As the “semi” denotes, however, this method does not explicitly form

*AA*

^{T}, which would be undesirable from the standpoint of numerical stability. Note that equations (21.5) are different from the equations

*R*

^{T}

*Rx*=

*A*

^{T}

*b*for an overdetermined least squares (LS) problem, where

*A*=

*Q*[

*R*

^{T}0]

^{T}∈ ℝ

^{m × n}with

*m*≥

*n*, which are also called seminormal equations (see §20.6).

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## Information & Authors

### Information

#### Published In

**ISBN (Print)**: 0898715210

**ISBN (Print)**: 978-0-89871-521-7

**ISBN (Online)**: 978-0-89871-802-7

#### Copyright

#### History

**Published online**: 23 March 2012

#### Keywords

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#### Keywords

- QA297 .H53 2002Numerical analysis—Data processing, Computer algorithms

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