topic Hi Baskin, in Intel® oneAPI Math Kernel Library

LU factorization of band matrix

baskin — Thu, 30 Oct 2014 09:41:22 GMT

I try to use https://software.intel.com/sites/products/documentation/doclib/iss/2013/mkl/mklman/GUID-03E2AA41-0886-485D-B0D7-1C2186119220.htm routine for band matrix factorization. I've wrote simple application that factorizes 4*4 matrix. However, I can't undertand relation between elements of matrix L and multipliers used during factorization m_i,j.Those mutipliers represent correct columns of matrix L, but order of elements m_i,j in the column is not correct. I assume, some permutation of rows should be performed based on ipiv vector.
Initial matrix:

-0.230 -6.980 0.000 0.000
2.540 2.460 2.560 0.000
-3.660 -2.730 2.460 -4.780
0.000 -2.130 4.070 -3.820

Result of dgbtrf:
0.000 0.000 0.000 -4.780
0.000 0.000 2.460 0.300
0.000 -2.730 -0.155 -3.292
-3.660 -6.808 4.254 -0.727
-0.694 -0.083 0.968 0.000
0.063 0.313 0.000 0.000

Permutation vector: 3 3 3 4

In this case
m₂₁ = -0.694 m₃₂ = -0.083 m₄₃= 0.968
m₃₁ = 0.063 m₄₂ = 0.313
Those values form matrix L_result:
1.000 0.000 0.000 0.000
-0.694 1.000 0.000 0.000
0.063 -0.083 1.000 0.000
0.000 0.313 0.968 1.000

However, the correct answer is L_correct:
1.000 0.000 0.000 0.000
0.063 1.000 0.000 0.000
-0.694 -0.083 1.000 0.000
0.000 0.313 0.968 1.000

Could you explain how L_result should be transformed into L_correct in general case?

Hi Baskin,

Ying_H_Intel — Mon, 03 Nov 2014 05:58:46 GMT

Hi Baskin,

Could you please attach your code and tell which platform you are working?

Best Regards,

Ying

Hi Ying,

baskin — Mon, 03 Nov 2014 18:17:32 GMT

Hi Ying,

Thank you for reply. I have solved my problem. I didn't understand output of gbtrf - the meaning of coefficients m_i,j. The explanation is pretty simple. LU factorization with partial pivoting requires swap of rows. For dense matrices, full rows are swapped( i.e columns from 1 to n). In gbtrf routine swap is performed using only "right" part of the row. For example, if you already found 10 columns, at step 11 you are going to swap rows 11 and 16(row 16 contains maximum element), then you should swap elements in row 11 starting from column 11, in row 16 starting from column 11 too. For dense matrix, you would swap elements starting from column 1.

Hi Baskin,

Ying_H_Intel — Tue, 04 Nov 2014 06:09:05 GMT

Hi Baskin,

Thanks for the updating and sharing the result.

Thanks

Ying