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From "Roman Werpachowski (JIRA)" <j...@apache.org>
Subject [jira] [Created] (MATH-1101) QR and Rank-revealing QR fail to find a least-squares solution
Date Fri, 21 Feb 2014 11:40:20 GMT
```Roman Werpachowski created MATH-1101:
----------------------------------------

Summary: QR and Rank-revealing QR fail to find a least-squares solution
Key: MATH-1101
URL: https://issues.apache.org/jira/browse/MATH-1101
Project: Commons Math
Issue Type: Bug
Affects Versions: 3.2
Reporter: Roman Werpachowski

QR and RRQR (rank-revealing) algorithms fail to find a least-squares solution in some cases.

The following code:

final RealMatrix A = new BlockRealMatrix(3, 3);
A.setEntry(0, 0, 1);
A.setEntry(0, 1, 6);
A.setEntry(0, 2, 4);
A.setEntry(1, 0, 2);
A.setEntry(1, 1, 4);
A.setEntry(1, 2, -1);
A.setEntry(2, 0, -1);
A.setEntry(2, 1, 2);
A.setEntry(2, 2, 5);
final RealVector b = new ArrayRealVector(new double[]{5, 6, 1});
final QRDecomposition qrDecomposition = new QRDecomposition(A);
final RRQRDecomposition rrqrDecomposition = new RRQRDecomposition(A);
final SingularValueDecomposition svd = new SingularValueDecomposition(A);
final RealVector xQR = qrDecomposition.getSolver().solve(b);
System.out.printf("QR solution: %s\n", xQR.toString());
final RealVector xRRQR = rrqrDecomposition.getSolver().solve(b);
System.out.printf("RRSQ solution: %s\n", xRRQR.toString());
final RealVector xSVD = svd.getSolver().solve(b);
System.out.printf("SVD solution: %s\n", xSVD.toString());

produces

QR solution: {-3,575,212,378,628,897; 1,462,586,882,166,368; -1,300,077,228,592,326.5}
RRSQ solution: {5,200,308,914,369,308; -2,127,399,101,332,898; 1,891,021,423,407,021}
SVD solution: {0.5050344462; 1.0206677266; -0.2405935347}

Showing that QR and RRQR algorithms fail to find the least-squares solution. This can also
be verified by calculating the dot product between columns of A and A*x - b:

// x = xQR, xRRQR or xSVD
final RealVector r = A.operate(x).subtract(b);
for (int i = 0; i < x.getDimension(); ++i) {
final RealVector columnVector = A.getColumnVector(i);
assertEquals(name, 0.0, r.dotProduct(columnVector), tolerance);
}

Only SVD method passes this test with decent tolerance (1E-14 or so).

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