Podemos ter certeza de que todos esses sistemas usam a implementação de referência do BLAS&LAPACK, em vez do OpenBLAS ou, por exemplo, o Accelerate da Apple?

Como os erros foram acionados de várias linguagens e máquinas, suponho que sim, mas meu próximo passo é tentar acioná-los chamando as bibliotecas de referência diretamente.

EDIT: para ser preciso, Julia usa OpenBLAS, não tenho certeza sobre Python. O que quero dizer é que suspeito que o erro não seja causado pela versão do OpenBLAS, mas gostaria de confirmar essa suspeita.

Eu certamente ficaria satisfeito se o problema não estivesse relacionado ao OpenBLAS, mas o numpy é frequentemente/geralmente distribuído com o OpenBLAS nos bastidores hoje em dia, e pode haver, por exemplo, acúmulo de erros do uso de instruções FMA em seus kernels BLAS otimizados.

Certo, acredito que confirmei o problema usando a biblioteca LAPACK de referência. Eu obtive e compilei usando homebrew em macos ( brew install lapack ). Chamei então a função zgeev no lapack dylib de Julia, passando primeiro as matrizes do exemplo 2, que deu erro, e depois com o exemplo 3, que funcionou.

Para referência, usei este código para envolver a chamada para LAPACK em uma função Julia myggev! e, em seguida, fiz myggev!('N', 'V', A, B) com as matrizes relevantes. Código adaptado diretamente do stdlib.

Fiz alguns testes com isso e obtive os seguintes resultados:

• todos os testes no meu trabalho de máquina pessoal
• testes em uma máquina xeon falham quando vinculados à distribuição openblas (somente exemplo 1)
• os testes nessa mesma máquina xeon funcionam quando vinculados à referência LAPACK ou OpenBLAS compilados a partir da fonte.

sem uma máquina que possa reproduzi-lo com uma versão que eu possa compilar, não consigo depurar.
@pablosanjose você se importaria de fazer alguns testes para mim?

Esta essência contém um arquivo de teste fortran e uma versão editada do zhgeqz que imprime alguns logs. Se você puder clonar o LAPACK, substituir zhgeqz pelo da essência e executar o arquivo de teste, talvez consiga identificar o problema.

PS o comprimento da linha é um pouco longo, então você terá que compilar o arquivo de teste com algo assim
make lib blaslib
gfortran -ffree-line-length-none -ggdb3 -fcheck=bounds -o test test.f90 librefblas.a liblapack.a -lblas -llapack

@thijssteel qual versão é a distribution openblas no seu caso? (Embora haja um problema estranho com compilações DYNAMIC_ARCH feitas em hardware Sandybridge não funcionando corretamente no SkylakeX) Se isso não puder ser reproduzido com puro Reference-LAPACK/BLAS, acho que devemos transferir esse problema para o rastreador OpenBLAS.

OpenBLAS instalado com apt.

libopenblas-base/bionic,now 0.2.20+ds-4 amd64 [installed,automatic]
libopenblas-dev/bionic,now 0.2.20+ds-4 amd64 [installed]

Acho que este é apenas um problema muito específico com simetria ou underflow que desaparece com a menor alteração nos erros de arredondamento, portanto, ainda deve ser tratado aqui. Além disso, pablos conseguiu reproduzi-lo usando a referência LAPACK.

Ops. 0.2.20 tem cerca de três anos e meio, tem LAPACK 3.7.0, não tem suporte específico para AVX512 xeons, tem vários problemas conhecidos de segurança de threads, reuso de registro nos kernels de montagem etc que já foram corrigidos (ou pelo menos substituídos por novos)

@thijssteel obrigado por investigar isso. Eu clonei, compilei e executei o teste. Os resultados estão nesta essência .

Como devemos interpretar isso?

EDIT: Isso está no meu macbook pro sob macos, a propósito, o que chamei de MA no OP.

Deixe-me apenas mencionar que as matrizes que postei não são ajustadas ou raras em nenhum sentido. Exemplos semelhantes surgem o tempo todo para mim em minha aplicação quando eu varro os parâmetros do meu modelo, então o problema é um problema muito real para mim, não acadêmico. É crucial, claro, verificar se isso é específico do OpenBLAS ou não.

Vejo que a essência está cortada. Vamos ver se consigo linká-lo aqui diretamente:
resultado.txt.zip

Eu detectei o problema.

Para resolver falhas de convergência (na maioria das vezes devido a simetrias), o LAPACK geralmente seleciona algum deslocamento aleatório para permutar o espectro após algumas iterações com falha. (veja a linha 370 da essência). Como esse deslocamento é quase zero, ele preservará a simetria. Eventualmente, por força bruta, obtemos convergência, mas são necessárias muitas iterações e eventualmente falhamos em resolver o lápis dentro do limite de iteração. Uma solução rápida é adicionar estratégias de mudança mais excepcionais (o algoritmo QR tem várias, por exemplo).

Também confirmei que isso também acontece ao compilar o OpenBLAS 0.2.20 da fonte. Uma solução rápida parece resolver o problema para mim. Atualizei a essência com a correção e farei um PR quando fizer mais alguns testes.

obrigado por relatar o problema com tantos detalhes e executar os testes para mim

Fantástico, obrigado!! Aguardo a correção

Depois de muita luta, e muita ajuda dos gurus da Julia, consegui compilar a biblioteca OpenBLAS com os patches da Julia mais o patch do #477, e voilá, as matrizes de exemplo 1, 2 e 3 não dão mais erros!! Mas, infelizmente, logo encontrei outras matrizes para as quais a correção não funcionou. Posto aqui outro exemplo que falha no meu Macbook Pro (máquina MA no OP), mesmo depois do #477

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Mesmo tipo de erro de antes.

EDIT: Eu incluí um segundo exemplo

Adaptei seu programa test.f90 em Fortran para esses novos exemplos. posto aqui os resultados

resultado2.txt.zip

teste2.f90.zip

EDIT: ah, desculpe, isso é sem #477. Deixe-me ver se consigo corrigi-lo e executar novamente.
EDIT2: Não tenho certeza se transferi as matrizes Julia corretamente para o Fortran, porque não consigo reproduzir o problema agora com test2.f90. Vou relatar se eu fizer progressos.

eu recebo

 RESULTS OF PENCIL            1  :            8
 RESULTS OF PENCIL            2  :           34

com o atual ramo develop do OpenBLAS que tem #477 incluído

Meu mal, você está certo. Eu não tinha compilado o patch #477 ao usar test2. Ele ainda falha em Julia, no entanto, com o OpenBLAS (espero que corrigido corretamente). Preciso investigar mais a fundo, posso ter convertido as matrizes para o Fortran errado.
Apenas uma verificação, @martin-frbg, você usou test2.f90 ou realmente copiou as matrizes A4,B4, A5, B5 do código Julia acima?

Ah, acabei de usar seu test2.f90. Não tenho certeza se consigo colocar Julia aqui a tempo.

Acho que precisa ser dito: aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaarg

Julian compartilhou um pouco da história deste problema no PR. Vou verificar se algum desses resolve o problema.

nota lateral, devemos adicionar esses lápis aos testes de unidade

OK. Apenas vamos tentar chegar ao fundo disso antes de mesclar. Não tenho 100% de certeza de que compilei o OpenBLAS com #477 corretamente, então não tenho certeza se estou soando o alarme cedo demais ou não.

Para ser claro, o que eu fiz:

• Eu peguei o diff #477 e o adaptei em um patch do OpenBLAS
• Eu compilei o OpenBLAS 0.3.13 com os patches da Julia mais o patch #477 usando o BinaryBuilder da Julia. Isso produziu um dylib que posso chamar de Julia
• Confirmei que os exemplos 1 e 2 falham no Julia 1.7 (que vem com o OpenBLAS 0.3.10 ) no meu macbook (MA), mas que eles não falharam ao chamar o zggev no dylib 0.3.13 corrigido de Julia
• Eu executei meu pacote, corrigido para chamar o dylib 0.3.13+#477 até que ele tropeçou mais uma vez com as matrizes 4 e 5 acima.

Há uma série de armadilhas potenciais complicadas aqui. Pode ser que eu não apliquei corretamente o patch #477 em 0.3.13 (primeira vez que tento algo remotamente assim, e o processo BinaryBuilder é bastante opaco para mim), e que 0.3.13 (ainda não empacotado com Jullia) sem o patch #477 não falha em 1, 2 e 3, mas sim em 4 e 5 (possibilidade remota). E esse 0.3.13+#477, aplicado corretamente, não falha de jeito nenhum (precisa reproduzir usando test2.f90, realmente). Também pode ser que as falhas sejam dependentes do sistema e que não falhem na máquina do @martin-frbg. De qualquer forma, vamos continuar cavando um pouco até esclarecermos.

@thijssteel , seria possível que você produzisse um test2.f90 a partir dos exemplos 4 e 5 acima, assim como fez para 1,2 e 3? Receio ter transferido as matrizes incorretamente.

477 deve ser aplicado de forma limpa ao OpenBLAS (em lapack-netlib). E eu (acho que) acabei de reproduzir o problema com a forma python do caso de teste 4 no que seria sua configuração "MB2" (desde que a ordenação da matriz seja a mesma para numpy e julia, que foi minha impressão dos primeiros casos de teste. basicamente apenas apimentado o caso 4 com muitas vírgulas e transformou todos os pontos e vírgulas em pares de colchetes)

Sim, isso é exatamente o que eu fiz para 1, 2 e 3. Então você confirmou a falha 4 no Linux+Python2+#477, sim? 4 não falha no Python3 + # 477 ou você não tentou? E o exemplo 5?

Linux, Python3, #477 foi o que eu tentei. Ainda não tentei o exemplo 5, pois me cansei de digitar vírgulas (e não teria uma solução elegante para o problema de qualquer maneira).

Você pode achar a seguinte função de oitava útil para isso (basta remover uma vírgula no final)

function [] = printasfortran (A,name)

[m,n] = size(A);

fprintf( '%s = reshape((/ ', name )
for j=1:n
  for i=1:m
    fprintf( "( %e, %e ),", A(i,j), 0.0 )
  endfor
endfor
fprintf( ' /), shape(%s))\n', name )

endfunction

A versão python de 4 e 5:

A4 = np.array([[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], [ 1.7391668762048442, -1.309613611600033, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.150333752409688, -2.619227223200066, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ -1.3096136116000332, 1.739166876204844, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -2.6192272232000664, 2.150333752409688, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 1.739166876204844, 0.0, 0.0, -1.3096136116000332, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 2.150333752409688, 0.0, 0.0, -2.6192272232000664, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 1.739166876204844, 0.0, 0.0, 0.0, 0.0, -1.3096136116000332, 0.0, -1.0, 0.0, 0.0, 2.150333752409688, 0.0, 0.0, 0.0, 0.0, -2.6192272232000664, 0.0], [ 0.0, 0.0, 0.0, 0.0, 1.7391668762048442, 0.0, 0.0, 0.0, 0.0, -1.309613611600033, 0.0, 0.0, 0.0, 0.0, 2.150333752409688, -1.0, 0.0, 0.0, 0.0, -2.619227223200066], [ 0.0, 0.0, -1.309613611600033, 0.0, 0.0, 1.7391668762048442, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -2.619227223200066, 0.0, -1.0, 2.150333752409688, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.739166876204844, -1.3096136116000332, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.150333752409688, -2.6192272232000664, 0.0, -1.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.309613611600033, 1.7391668762048442, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -2.619227223200066, 2.150333752409688, -1.0, 0.0], [ 0.0, 0.0, 0.0, -1.309613611600033, 0.0, 0.0, 0.0, 0.0, 1.7391668762048442, 0.0, 0.0, 0.0, 0.0, -2.619227223200066, 0.0, 0.0, 0.0, -1.0, 2.150333752409688, 0.0], [ 0.0, 0.0, 0.0, 0.0, -1.3096136116000332, 0.0, 0.0, 0.0, 0.0, 1.739166876204844, 0.0, 0.0, 0.0, 0.0, -2.6192272232000664, 0.0, -1.0, 0.0, 0.0, 2.150333752409688]]) + 0.0j

B4 = np.array([[1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.7391668762048442, 1.3096136116000332, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.309613611600033, -1.739166876204844, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.739166876204844, 0.0, 0.0, 1.309613611600033, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.739166876204844, 0.0, 0.0, 0.0, 0.0, 1.309613611600033, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.7391668762048442, 0.0, 0.0, 0.0, 0.0, 1.3096136116000332], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.3096136116000332, 0.0, 0.0, -1.7391668762048442, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.739166876204844, 1.309613611600033, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.3096136116000332, -1.7391668762048442, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.3096136116000332, 0.0, 0.0, 0.0, 0.0, -1.7391668762048442, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.309613611600033, 0.0, 0.0, 0.0, 0.0, -1.739166876204844]]) + 0.0j

A5 = np.array([[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 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Para referência, mudei um pouco da história para #102

@thijssteel : como o #421 se sai com esses lápis? Você sabe?

DLAQZ0 adiará o cálculo para DHGEQZ para esses lápis (n < 75)

É de se esperar que se saia pelo menos um pouco melhor de certa forma, porque o AED pode resolver muitos desses problemas.
Também sei que será pelo menos melhor do que a maioria das outras implementações por causa de alguma proteção para estouro no cálculo de turno duplo.

Por outro lado, sei que o DLAQR0 faz muito mais para resolver problemas de convergência. Varia os tamanhos das janelas do AED, tem turnos excepcionais mais avançados, ... se o DLAQZ0 tiver problemas de convergência, eu poderia adicionar essas técnicas

Eu usei a função oitava do @thijssteel para escrever as matrizes corretamente. Este é o resultado que obtenho sob macos com o patch #477.

Eu acho que eles mostram problemas de convergência com o exemplo 4. Estou lendo isso corretamente?

teste2.f90.zip
resultado2.txt.zip

@thijssteel : como o #421 se sai com esses lápis? Você sabe?

Extravagante. O nº 421 está relacionado a https://arxiv.org/abs/2007.03576 ?

Não, para https://arxiv.org/pdf/1902.10954.pdf (mas esse artigo é mais sobre uma extensão teórica do QZ, não sobre essa implementação especificamente).
Tenho certeza que o código de Mirko supera o meu.

Isso provavelmente é óbvio para você, mas uma coisa comum que vejo com todas essas matrizes é que elas têm muitos valores singulares que são iguais à precisão da máquina. Acho que seu comentário sobre simetrias é exatamente isso.

Como lápis pequenos são mais desejáveis ​​para testes unitários, posto aqui um exemplo 12x12 que falha após #477 para mim

A6 = ComplexF64[0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0; 3.507883011020636 4.043337864077769 0.0 0.0 0.0 0.0 7.883337105374606 8.086675728155537 0.0 -1.0 0.0 0.0; 4.043337864077769 3.5078830110206356 0.0 0.0 0.0 0.0 8.086675728155537 7.883337105374604 -1.0 0.0 0.0 0.0; 0.0 0.0 3.5078830110206356 0.0 0.0 4.043337864077769 0.0 -1.0 7.883337105374604 0.0 0.0 8.086675728155537; 0.0 0.0 0.0 3.5078830110206356 4.043337864077769 0.0 -1.0 0.0 0.0 7.883337105374604 8.086675728155537 0.0; 0.0 0.0 0.0 4.043337864077769 3.507883011020636 0.0 0.0 0.0 0.0 8.086675728155537 7.883337105374606 -1.0; 0.0 0.0 4.043337864077769 0.0 0.0 3.507883011020636 0.0 0.0 8.086675728155537 0.0 -1.0 7.883337105374606];
B6 = ComplexF64[1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 -3.507883011020636 -4.043337864077769 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 -4.043337864077769 -3.5078830110206356 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -3.5078830110206356 0.0 0.0 -4.043337864077769; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -3.5078830110206356 -4.043337864077769 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.043337864077769 -3.507883011020636 0.0; 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.043337864077769 0.0 0.0 -3.507883011020636];

Isso provavelmente é óbvio para você, mas uma coisa comum que vejo com todas essas matrizes é que elas têm muitos valores singulares que são iguais à precisão da máquina. Acho que seu comentário sobre simetrias é exatamente isso.

Hmmm, a ligação entre valores singulares e autovalores não é trivial, pelo menos para mim.

A simetria é mais sobre 2 autovalores sendo igualmente próximos ao deslocamento. Então eles convergirão igualmente rápido. A mudança não converge para um único autovalor e, eventualmente, nunca obtemos convergência. Isso é bastante comum em matrizes reais porque os autovalores são simétricos em relação ao eixo real.

Outras causas comuns são under/overflow no cálculo da coluna de mudança ou deflações vigilantes autodestrutivas raras.

O problema que enfrentamos agora é que, para perturbar a simetria e a falta de convergência, queremos apenas fazer uma mudança um tanto aleatória. Mas apenas escolher algo aleatório nem sempre funciona, porque precisa estar significativamente mais próximo de um autovalor do que do outro. Então, de alguma forma, nos baseamos nas entradas inferiores, mas elas podem, em casos raros, ser zero, ou ser alguma combinação linear do traço/determinante do sub-lápis inferior, que novamente possui propriedades especiais que podem prejudicar seu efeito para alguns lápis...

Então sim, enquanto eu gostaria de agir que eu vou inventar alguma estratégia genial com muitas derivações. Vou tentar um monte de coisas aleatórias e ver o que funciona. Possivelmente vou tirar a poeira do meu número favorito: 1302. Obrigado por todos os lápis, me dá muitos testes para trabalhar.

Obrigado pela informação. Se você encontrar mais correções em potencial, vou experimentá-las no meu código. Embora não seja o comentário mais construtivo, noto que não consegui fazer o MKL da Intel falhar uma única vez (eu tentei!), então deve existir alguma estratégia ideal para resolver isso que eles usam.

Aqui está um pedaço de código que nos foi fornecido pelo MathWorks há algum tempo. Tivemos problemas com esse código em nosso conjunto de testes, então nunca o integramos. Meu entendimento é que este código fez o seu caminho nos códigos MKL e MathWorks LAPACK. Se alguém puder tentar brincar com isso e ver se isso ajuda, isso seria ótimo.

Veja: http://math.ucdenver.edu/~langou/zhgeqz--mathworks.f
E uma diferença entre o código MathWorks (à esquerda) e a corrente do LAPACK (à direita) está em:
http://math.ucdenver.edu/~langou/Diff.html

Estou tentando confirmar com o MathWorks se o código que estou fornecendo acima é o correto. Eu não tenho 100% de certeza. Mas se alguém puder experimentar o código e ver se isso ajuda nesses lápis, isso seria útil.

Ok, então sim. Que tal esquecermos toda a matemática sofisticada que eu estava explicando...

As linhas 722-732 não calculam corretamente $AB^{-1}$. Mais detalhadamente, AB12 nunca é calculado, em vez disso, o código usa AD12.

Depois de corrigir isso (e para uma boa medida, atualizando um pouco o cálculo de deslocamento), a estratégia de deslocamento excepcional não importa mais para os exemplos de Pablo porque obtemos convergência regular dentro do limite de iteração.

Atualizei a essência se o Pablo quiser confirmar, mas parece improvável que a dependência da máquina ainda seja um problema.

É também provável que o MKL e o trabalho de matemática tenham um desempenho tão bom

Fico feliz em informar que com a nova correção não consigo mais encontrar lápis não convergentes com meu código. Muito bem @thijssteel !! Obrigado a todos

EDIT: para ser claro, testei este commit do #477. O seguinte afeta apenas chgeqz.f, mas estou chamando a versão dupla complexa, então acho que não afeta meus testes.