# \[ ruby-Bugs-9360 \] Matrix inverse problem

**URL:** <https://www.ruby-forum.com/t/ruby-bugs-9360-matrix-inverse-problem/89268>\
**Category:** Development\
**Created:** [March 19, 2007, 5:08am UTC](https://www.ruby-forum.com/t/ruby-bugs-9360-matrix-inverse-problem/89268 "2007-03-19T05:08:52Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![Yukihiro\_M](https://www.ruby-forum.com/letter_avatar_proxy/v4/letter/y/c57346/32.png) [@Yukihiro\_M](https://www.ruby-forum.com/u/Yukihiro_M)\
**Post date:** [March 19, 2007, 5:08am UTC](https://www.ruby-forum.com/t/ruby-bugs-9360-matrix-inverse-problem/89268/1 "2007-03-19T05:08:52Z")

</div>

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rubyforgee$B%P%0%l%]!\<%H$K0J2\<$N$h$&$J%l%]!\<%H$,Mh$F$^$9$,!";de(B  
e$B$K$OH=CG$G$-$^$;$s!#$I$&$G$7$g$&!#@PDM$5$s!)e(B

# 

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# e$B$G$9!#e(B

* * *

There is a problem with the Matrix.inverse method.  
Given the following matrix

```
m = Matrix[[0.417787968829298, 0.542037605627772, 

```

0.729142268138943],  
[0.0, 6.12303176911189e-17, 1.0],  
[-0.542037605627772, 0.417787968829298, -2.55812900589452e-17]]

the result of

```
m.inverse * m

```

is

```
[[1.0, 0.0, 0.650423523448542],
[5.55111512312578e-17, 1.0, 0.84385869292008],
[0.0, 9.24446373305873e-33, 1.0]]

```

but it should be

```
[[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 1.0]]

```

or close to this!

* * *

The patch below should “fix” the inverse method. It adds partial  
pivoting to the Gauss-Jordan algorithm, making it stable. (I used Ruby  
1.8.5 (2006-08-22).)

I don’t think there is an error in the algorithm, and the matrix is not  
ill-conditioned (a condition number of about 2.5 isn’t so bad).

Looks to me like BigDecimal uses Gaussian elimination with partial  
pivoting too, or at least that’s what the comments say. It includes some  
additional scaling, but this may be specific to BigDecimal (precision  
works differently for BigDecimals and floats).

Peter

— lib/matrix.rb.old 2007-03-19 01:46:44.000000000 +0100  
+++ lib/matrix.rb 2007-03-19 03:26:38.000000000 +0100  
@@ -590,15 +590,21 @@  
a = src.to\_a

```
 for k in 0..size

```

- 

```
 if (akk = a[k][k]) == 0

```

- 

```
   i = k

```

- 

```
   begin

```

- 

```
     Matrix.Raise ErrNotRegular if (i += 1) > size

```

- 

```
   end while a[i][k] == 0

```

- 

```
 i = k

```

- 

```
 akk = a[k][k].abs

```

- 

```
 for j in (k+1)..size

```

- 

```
   v = a[j][k].abs

```

- 

```
   if v > akk

```

- 

```
     i = j

```

- 

```
     akk = v

```

- 

```
   end

```

- 

```
 end

```

- 

```
 Matrix.Raise ErrNotRegular if akk == 0

```

- 

```
 if i != k
   a[i], a[k] = a[k], a[i]
   @rows[i], @rows[k] = @rows[k], @rows[i]

```

- 

```
   akk = a[k][k]
 end

```

- 

```
 akk = a[k][k]

 for i in 0 .. size
   next if i == k
```

---

<div class="post-metadata">

**Author:** ![Yukihiro\_M](https://www.ruby-forum.com/letter_avatar_proxy/v4/letter/y/c57346/32.png) [@Yukihiro\_M](https://www.ruby-forum.com/u/Yukihiro_M)\
**Post date:** [March 19, 2007, 5:55am UTC](https://www.ruby-forum.com/t/ruby-bugs-9360-matrix-inverse-problem/89268/2 "2007-03-19T05:55:56Z")

</div>

e$B$1$$$8$e!w$$$7$D$+$G$9e(B.

In [ruby-dev:30627] the message: “[ruby-dev:30627] [ruby-Bugs-9360]  
Matrix inverse problem”, on Mar/19 13:08(JST) Yukihiro M.  
writes:

> e$B$^$D$b$He(B e$B$f$-$R$m$G$9e(B
> 
> rubyforgee$B%P%0%l%]!\<%H$K0J2\<$N$h$&$J%l%]!\<%H$,Mh$F$^$9$,!";de(B  
> e$B$K$OH=CG$G$-$^$;$s!#$I$&$G$7$g$&!#@PDM$5$s!)e(B

e$B5W$7$V$j$K?tCM2r@O$NK$rD4$Y$^$7$?e(B.  
e$B$3$N$h$&$J=hM}$O\>o\<1$N$h$&$G$9e(B(e$BItJ,E\*%T%%C%F%#%s%0$H8@$&$i$7$$e(B).

e$B7W;;%3%9%H$bL5;k$G$-$k$0$i$$7ZHy$H$J$C$F$$$k$N$Ge(B,  
e$B$3$A$i$KJQ$($FNI$$$He(B  
e$B;W$$$^$9e(B.

cie$B$O\>\>K$5$s$NJ}$G$7$^$9e(B?

\_\_  
----------------------------------------------------\>\> e$B@PDMe(B  
e$B7=\<ye(B \<\<—  
----------------------------------\>\> e-mail: removed\_email\_address@domain.invalid \<\<—

---

<div class="post-metadata">

**Author:** ![Yukihiro\_M](https://www.ruby-forum.com/letter_avatar_proxy/v4/letter/y/c57346/32.png) [@Yukihiro\_M](https://www.ruby-forum.com/u/Yukihiro_M)\
**Post date:** [March 19, 2007, 6:11am UTC](https://www.ruby-forum.com/t/ruby-bugs-9360-matrix-inverse-problem/89268/3 "2007-03-19T06:11:57Z")

</div>

e$B$^$D$b$He(B e$B$f$-$R$m$G$9e(B

In message “Re: [ruby-dev:30632] Re: [ruby-Bugs-9360] Matrix inverse  
problem”  
on Mon, 19 Mar 2007 13:54:43 +0900, removed\_email\_address@domain.invalid  
(e$B@PDM7=\<ye(B) writes:

|e$B5W$7$V$j$K?tCM2r@O$NK$rD4$Y$^$7$?e(B.  
|e$B$3$N$h$&$J=hM}$O\>o\<1$N$h$&$G$9e(B(e$BItJ,E\*%T%%C%F%#%s%0$H8@$&$i$7$$e(B).  
|  
|e$B7W;;%3%9%H$bL5;k$G$-$k$0$i$$7ZHy$H$J$C$F$$$k$N$Ge(B, e$B$3$A$i$KJQ$($FNI$$$He(B  
|e$B;W$$$^$9e(B.

e$BN;2r$7$^$7$?!#e(B

|cie$B$O\>\>K$5$s$NJ}$G$7$^$9e(B?

trunke$B$O;d$G$d$j$^$9!#e(B1.8e$B$Oe(Bknue$B$5$s$K$\*G$$;$7$^$9!#e(B

---

<div class="post-metadata">

**Author:** ![Yukihiro\_M](https://www.ruby-forum.com/letter_avatar_proxy/v4/letter/y/c57346/32.png) [@Yukihiro\_M](https://www.ruby-forum.com/u/Yukihiro_M)\
**Post date:** [March 19, 2007, 5:29pm UTC](https://www.ruby-forum.com/t/ruby-bugs-9360-matrix-inverse-problem/89268/4 "2007-03-19T17:29:24Z")

</div>

At Mon, 19 Mar 2007 14:11:28 +0900,  
matz wrote:

> |ciは松本さんの方でします?
> 
> trunkは私でやります。1.8はknuさんにお任せします。

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