# Improve numerical stability for method tallSkinnyQR.

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#### Details

• Improvement
• Status: Resolved
• Minor
• Resolution: Won't Fix
• 2.2.0
• None

#### Description

In method tallSkinnyQR, the final Q is calculated by A * inv(R) (Github Link). When the upper triangular matrix R is ill-conditioned, computing the inverse of R can result in catastrophic cancellation. Instead, we should consider using a forward solve for solving Q such that Q * R = A.

I first create a 4 by 4 RowMatrix A = (1,1,1,1;0,1E-5,0,0;0,0,1E-10,1;0,0,0,1E-14), and then I apply method tallSkinnyQR to A to find RowMatrix Q and Matrix R such that A = Q*R. In this case, A is ill-conditioned and so is R.

See codes in Spark Shell:

```import org.apache.spark.mllib.linalg.{Matrices, Vector, Vectors}
import org.apache.spark.mllib.linalg.distributed.RowMatrix

// Create RowMatrix A.
val mat = Seq(Vectors.dense(1,1,1,1), Vectors.dense(0, 1E-5, 1,1), Vectors.dense(0,0,1E-10,1), Vectors.dense(0,0,0,1E-14))
val denseMat = new RowMatrix(sc.parallelize(mat, 2))

// Apply tallSkinnyQR to A.
val result = denseMat.tallSkinnyQR(true)

// Print the calculated Q and R.
result.Q.rows.collect.foreach(println)
result.R

// Calculate Q*R. Ideally, this should be close to A.
val reconstruct = result.Q.multiply(result.R)
reconstruct.rows.collect.foreach(println)

// Calculate Q'*Q. Ideally, this should be close to the identity matrix.
result.Q.computeGramianMatrix()

System.exit(0)
```

it will output the following results:

```scala> result.Q.rows.collect.foreach(println)
[1.0,0.0,0.0,1.5416524685312E13]
[0.0,0.9999999999999999,0.0,8011776.0]
[0.0,0.0,1.0,0.0]
[0.0,0.0,0.0,1.0]

scala> result.R
1.0  1.0     1.0      1.0
0.0  1.0E-5  1.0      1.0
0.0  0.0     1.0E-10  1.0
0.0  0.0     0.0      1.0E-14

scala> reconstruct.rows.collect.foreach(println)
[1.0,1.0,1.0,1.15416524685312]
[0.0,9.999999999999999E-6,0.9999999999999999,1.00000008011776]
[0.0,0.0,1.0E-10,1.0]
[0.0,0.0,0.0,1.0E-14]

scala> result.Q.computeGramianMatrix()
1.0                 0.0                 0.0  1.5416524685312E13
0.0                 0.9999999999999998  0.0  8011775.999999999
0.0                 0.0                 1.0  0.0
1.5416524685312E13  8011775.999999999   0.0  2.3766923337289844E26
```

With forward solve for solving Q such that Q * R = A rather than computing the inverse of R, it will output the following results instead:

```scala> result.Q.rows.collect.foreach(println)
[1.0,0.0,0.0,0.0]
[0.0,1.0,0.0,0.0]
[0.0,0.0,1.0,0.0]
[0.0,0.0,0.0,1.0]

scala> result.R
1.0  1.0     1.0      1.0
0.0  1.0E-5  1.0      1.0
0.0  0.0     1.0E-10  1.0
0.0  0.0     0.0      1.0E-14

scala> reconstruct.rows.collect.foreach(println)
[1.0,1.0,1.0,1.0]
[0.0,1.0E-5,1.0,1.0]
[0.0,0.0,1.0E-10,1.0]
[0.0,0.0,0.0,1.0E-14]

scala> result.Q.computeGramianMatrix()
1.0  0.0  0.0  0.0
0.0  1.0  0.0  0.0
0.0  0.0  1.0  0.0
0.0  0.0  0.0  1.0
```

#### People

Unassigned
Huamin Li