# MATLAB and LAPACK implementation of the SVD algorithm - not the same output?

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Danijel Domazet il 22 Feb 2022
Modificato: David Goodmanson il 23 Feb 2022
I am using the LAPACK C library's singular value decomposition function
LAPACKE_cgesvd()
and trying to get the same result as MATLAB's svd() function (in case of complex input).
But the outputs are not the same.
Here is one example:
in = [1+2i 2+4i 3+6i 4+8i;
2+3i 4+6i 6+9i 8+12i;
3+4i 6+8i 9+12i 12+16i;
4+5i 8+10i 12+15i 16+20i];
[U,S,V] = svd(in);
This gives
U =
-0.109108945117997 - 0.218217890235993i 0.909365643461770 + 0.318130360459734i 0.036386779676407 - 0.084386315028456i 0.052751601014692 + 0.033100021335301i
-0.218217890235992 - 0.327326835353989i -0.037388878899514 - 0.059582067281587i -0.116960851562535 + 0.558535127042627i -0.249442499023482 + 0.672627129227888i
-0.327326835353989 - 0.436435780471985i -0.117020313474447 + 0.048160693713279i -0.345516960640487 + 0.358878427669969i 0.391668855331377 - 0.533654905367364i
-0.436435780471985 - 0.545544725589981i -0.219753961051779 - 0.050933678992021i 0.293035065226691 - 0.576080183457497i -0.205991407911317 + 0.029126130759720i
S =
50.199601592044544 0 0 0
0 0.000000000000004 0 0
0 0 0.000000000000000 0
0 0 0 0.000000000000000
V =
-0.182574185835055 + 0.000000000000000i -0.749839996855303 + 0.000000000000000i 0.635929749093960 + 0.000000000000000i 0.000000000000000 + 0.000000000000000i
-0.365148371670110 + 0.000000000000000i -0.099644835021179 - 0.055294049648946i -0.222326993896782 - 0.065198538162360i -0.274861567584794 - 0.851146943050864i
-0.547722557505166 + 0.000000000000000i 0.582096115689033 + 0.184313498829819i 0.529113396624621 + 0.217328460541199i -0.000000000000000 + 0.000000000000001i
-0.730296743340221 + 0.000000000000000i -0.199289670042359 - 0.110588099297891i -0.444653987793565 - 0.130397076324720i 0.137430783792397 + 0.425573471525431i
When I run the LAPACK C code, I get:
U
[0][0] = -0.109108955 -0.218217865i [0][1] = +0.719114125 -0.452113479i [0][2] = +0.022277016 +0.064515218i [0][3] = -0.409647375 +0.215580061i
[1][0] = -0.218217909 -0.327326864i [1][1] = +0.084657431 -0.228100479i [1][2] = +0.500972092 -0.310676992i [1][3] = +0.300640881 -0.590053499i
[2][0] = -0.327326834 -0.436435789i [2][1] = +0.109215073 +0.273434073i [2][2] = -0.344847411 -0.481100261i [2][3] = +0.325270653 +0.399385184i
[3][0] = -0.436435819 -0.545544684i [3][1] = -0.340743810 +0.128338531i [3][2] = +0.002677787 +0.545402348i [3][3] = -0.279374570 -0.061697662i
S
[0] = 50.1995964 [1] = 1.69897112e-06 [2] = 1.54533879e-07 [3] = 3.96336745e-14
Vtransposed
[0][0] = -0.182574213 -0.000000000i [0][1] = -0.365148425 -0.000000000i [0][2] = -0.547722638 +0.000000028i [0][3] = -0.730296850 -0.000000000i
[1][0] = -0.219238073 -0.000000000i [1][1] = +0.210008949 +0.144917369i [1][2] = -0.626949847 -0.483057886i [1][3] = +0.420017421 +0.289834768i
[2][0] = -0.958436847 -0.000000000i [2][1] = +0.021519713 -0.033149216i [2][2] = +0.247748464 +0.110497266i [2][3] = +0.043038044 -0.066298351i
[3][0] = -0.000000639 -0.000000000i [3][1] = -0.892759502 +0.054592617i [3][2] = +0.000000003 -0.000000026i [3][3] = +0.446379900 -0.027296286i
U: Seems like the first U column is good! But the rest is totally different.
S: All good, if we approximate small C outputs to be zero.
Vt: Only the first element seems to be the same, the rest doesn't match to Matlab's output.
What is the reason for this?
Am I doing something wrong?
##### 8 CommentiMostra 6 commenti meno recentiNascondi 6 commenti meno recenti
Danijel Domazet il 22 Feb 2022
Modificato: Danijel Domazet il 22 Feb 2022
Let's put it this way: is there an input for which LAPACK and MATLAB would give the same result?
Or, what's the best try?
Danijel Domazet il 22 Feb 2022
Modificato: Danijel Domazet il 22 Feb 2022
OK, I used rand(4,4) + i*rand(4,4), and the result is much better:
in = ...
[0.814723686393179 + 0.421761282626275i 0.63235924622541 + 0.655740699156587i ...
0.957506835434298 + 0.678735154857773i 0.957166948242946 + 0.655477890177557i ...
;
0.905791937075619 + 0.915735525189067i 0.0975404049994095 + 0.0357116785741896i ...
0.964888535199277 + 0.757740130578333i 0.485375648722841 + 0.171186687811562i ...
;
0.126986816293506 + 0.792207329559554i 0.278498218867048 + 0.849129305868777i ...
0.157613081677548 + 0.743132468124916i 0.8002804688888 + 0.706046088019609i ...
;
0.913375856139019 + 0.959492426392903i 0.546881519204984 + 0.933993247757551i ...
0.970592781760616 + 0.392227019534168i 0.141886338627215 + 0.0318328463774207i ...
];
Result is:
U = ...
[-0.42312740984401409 -0.35959265421087588i
0.13117268904541174 +0.44787859162628668i ...
0.11250427785751983 -0.24706940551578091i
-0.38438224614546779 -0.50239884161662351i ...
-0.33759289967935291 -0.32429926333442366i
-0.06936700508106322 -0.47809332716460873i ...
0.62193638822998987 -0.24622726102506251i
0.02217454979080485 +0.31551793178629584i ...
-0.12989079970942036 -0.439785223293499i
0.30358385553377837 +0.4177223468250193i ...
-0.25124960829083876 -0.062014277681078311i
0.42385988499316413 +0.52576884915022781i ...
-0.38050005142283305 -0.34271619213843124i
-0.47813520137109283 -0.23139813043816407i ...
-0.46987090655301544 +0.43716810967614084i
-0.19832860352154408 +0.066167190021447483i ...
];
The LAPACK gives:
[0][0] = -0.423127532 -0.359592706i
[0][1] = +0.131172717 +0.447878778i
[0][2] = -0.112504244 +0.247069359i
[0][3] = +0.384382606 +0.502398610i
[1][0] = -0.337592900 -0.324299276i
[1][1] = -0.069367193 -0.478093356i
[1][2] = -0.621936560 +0.246227175i
[1][3] = -0.022174668 -0.315517843i
[2][0] = -0.129890800 -0.439785272i
[2][1] = +0.303584039 +0.417722374i
[2][2] = +0.251249373 +0.062014218i
[2][3] = -0.423860222 -0.525768697i
[3][0] = -0.380500108 -0.342716217i
[3][1] = -0.478135169 -0.231398359i
[3][2] = +0.469871104 -0.437167943i
[3][3] = +0.198328614 -0.066167258i
The results are OK, or very close, it's only the sign of the last two rows that are different. Why would the signs be different? I would hope it will not matter in my further calculations...

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### Risposte (1)

David Goodmanson il 23 Feb 2022
Modificato: David Goodmanson il 23 Feb 2022
Hi Danijel,
actually it's the columns that have different signs. That's for the following reason: Suppose the svd of the matrix 'in' is
in = USV'
and let
A =
[1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 -1]
Then, since A*A = I (4x4 identity)
in = U(AA)S(AA)V'.
Since S is diagonal, ASA = S and
in = (UA)S(AV')
So if U is a solution, so is UA. But UA just multiplies the 4th column of U by a minus sign. AV' is the accomanying solution and it multiplies the 4th row of V" by a minus sign, which means the 4th column of V gets a minus sign. You can obviously do the same for any column of both U and V, meaning that the signs of the columns are arbitrary. In your case the 3rd and 4th columns of U have the minus sign compared to Lapack, so if you take a look at V, its 3rd and 4th columns should also have a minus sign compared to Lapack. Once the signs are set, you just have to stay consistent for the rest of the calculation.
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