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Una propuesta para el cálculo del pseudoespectro en unidades de procesamiento = gráfico

 

A proposal for pseudospectra computation on graphic processor units

 

Zenaida Natividad Castillo Marrero. [1], Gustavo Adolfo Colmenares Pacheco. [2], Paulina Elizabeth Valverde Aguirre. [3] &= amp; Víctor Oswaldo Cevallos Vique. [4]

 

Recibido: 09-03-2021 / Revisado: 16-03-2021 /Aceptado: 06-04-2021/ Publicado: 05-05-2= 021

 

Abstract.                         =                          DOI: https://doi= .org/10.33262/concienciadigital.v4i2.1.1703

 

Introduction. Computation of matrix pseudospectra is required = in many applications when modeling by differential equations. This computation= is really expensive, especially for large matrices, for which highly parallelizable algorithms have been successfully implemented on high performance computers. Objective. We present an exploratory analysis= of the pseudospectra computation in a hybrid architecture CPU-GPU where the graphics processing unit performs the massive parallel computation. Meth= odology. A proposal is formulated after analyzing some parallel implementations on h= igh performance computers, methods based on Krylov methods, and the capacities = of the graphics processor units for massive computation in the large-scale setting. Results. The proposal is attractive since the graphics processing units currently can be found on a wide range of computers, or ca= n be adapted to any computer at a very a low cost. Conclusions. In this document we describe a general scheme for the parallel computation of pseud= ospectra on a hybrid architecture CPU-GPU.

Keywords: Pseudospectra, Eigenvalues, Krylov methods, GPU, NVIDIA.

 

Resumen.

 

Introducción. = El cálculo del pseudoespectro de matrices es requerido en muchas aplicaciones modeladas por ecuaciones diferenciales y discretizadas en tiempo y espacio. Este cálculo resulta ser muy costoso computacionalmente, sobre todo para matrices de gran magnitud, para las cuales se han implementado con éxito métodos altamente paralelizables ejecutados en máquinas de alto rendimien= to. Objetivo. Se presenta un análisis exploratorio del cálculo del pseudoespectro y su = posible implementación en una arquitectura híbrida CPU-GPU, en la cual el cómputo masivo y paralelo se realice en las unidades de procesamiento gráfico. = Metodología. Para formular la propuesta se analizan algunas implementaciones de este cá= lculo que han resultado efectivas en máquinas de alto rendimientos, el uso de mÃ= ©todos basados en métodos de Krylov, y las capacidades de las unidades de procesamiento gráfico en el cómputo masivo. Resultados.  La propuesta resulta de interés debido= a que la mayor parte de este cálculo puede realizarse en unidades de procesamien= to gráfico, que en la actualidad son fáciles de adquirir a bajo costo, y est= án incluidas en la mayoría de las marcas y modelos presentes en el mercado. <= b>Conclusiones. En este documento se describe un esquema general para la paralelización del cálculo del pseudoespectro en una arquitectura híbrida CPU-GPU.

Palabras claves: Pseudoespectro, Autovectores, Métodos de Krylov, GPU, NVIDIA.

 

Introducción

En los modelos lineales = la investigación se basa en la información provista por los autovalores, y p= ara muchos problemas relacionados con la ciencia y la ingeniería este análisis es satisfactorio, sobre todo cuando las matrices son normales. Sin embargo, en algunas áreas de la industria, las matrices que se derivan del modelaje matemático, no poseen columnas ortogonales, y en caso de que puedan ortogonalizarse el proceso numérico es costoso o resulta en aproximaciones= muy pobres o poco valederas para la toma de decisiones. Adicionalmente, la no-normalidad influye negativamente en la convergencia de los métodos iterativos. En estos casos el pseudoespectro es una herramienta útil, que proporciona una alternativa visual para reforzar el análisis espectral. Ã= reas de aplicación exitosa de técnicas con autovalores y pseudoautovalores inc= luyen el estudio de la acústica, análisis estructural, mecánica cuántica, mec= ánica de fluidos, procesos de revestimiento de superficies, entre otros.<= /span>

Con la llegada de los ordenadores de alto rendimiento, que permiten cálculos en paralelo, se abr= ió la posibilidad de avanzar con el procesamiento masivo de datos en aplicaciones= a gran escala. De esta manera muchos autores se avocaron a la tarea de redefi= nir o adaptar sus algoritmos para sacar el mayor provecho al uso de estas tecnologías, entre ellos vale la pena mencionar los trabajos de Bekas et a= l. (2001), Mezher & Philippe (2011), y Noschese & Reichel (2015). Estos algoritmos han sido programados en su mayoría a través de una arquitectur= a de pase de mensajes como MPI o OpenMP, ver por ejemplo el trabajo de Bekas et = al. (2002) con MPI_Matlab o el de Minini et al. (2011) con un híbrido MPI-Open= MP, ambos implementados en computadores con decenas de miles de procesadores.

Por otra parte, en la ú= ltima década, las unidades de procesamiento gráfico o GPU (por sus siglas en in= glés) adaptadas a la gran mayoría de microcomputadores y computadores personales= , han probado tener una capacidad de cómputo que puede ser usada para aligerar significativamente la carga de la unidad principal de procesamiento o CPU. = Este hecho, sumado a que no siempre se dispone de una máquina de alto rendimien= to, nos lleva a considerar la implementación de estos métodos bajo un esquema híbrido que involucre cálculos numéricos densos en las GPU y manejo de entrada-salida a cargo del CPU.

En este trabajo se descr= ibe una propuesta para la implementación de un esquema paralelo para el cálcu= lo del pseudoespectro de matrices en un modelo CPU-GPU.<= /span>

Metodologia=

Para el desarrollo de la propuesta se describe el problema de hallar el pseudoespectro de matrices de gran tamaño y se mencionan algunos métodos que han resultado efectivos en= su resolución en computadores de alto rendimiento. En particular se detalla el funcionamiento de los algoritmos que integran la propuesta y su posible implementación en un modelo de arquitectura CPU-GPU.

El problema de los autovalore= s y el pseudoespectro

El espectro de una matriz es el conjunto de escalares z tales que existe un vector no nulo x tal que  . El escalar z, generalmente complejo, se conoce como autovalor y se dice que x es su autovector asociado. También pudiéramos decir que z es autovalor d= e A si y sólo si . Esto conlleva a la siguiente definición.

Definición 1: Dada una matriz , se define el espect= ro de A, denotándolo como , de = la siguiente manera:

Ahora bien, si entonces la matriz  =   no está defin= ida, y a tal efecto se define su norma como infinita ( ). Pero, ¿qué ocurre cuando  es finita pero muy grande?, la formulación de esta pregunta es la que permite construir la primera definición de pseudoespectro:

Definición 2: Dada una matriz  , se define el e-pseudoespectro de , para e > 0, denotándolo como , de la siguiente man= era:

Otra definición equival= ente, muy usada en teoría de perturbaciones, expresa al pseudoespectro en térmi= nos de los autovalores de una matriz perturbada en un épsilon= ( ).

Definición 3: Dada una matriz  , se define el e-pseudoespectro de , denotándolo como <= /span> , de la siguiente man= era:

Si <= ![if !msEquation]>  se encuentra en el pseudoes= pectro, entonces se denomina pseudoautovalor de <= ![if !msEquation]> . A cada pseudoautovalor <= ![if !msEquation]>  puede asociarse un pseudoautovector <= ![if !msEquation]> , en general, no único. Esto permite dar otra definición del pseudoespectro:

Definición 4: Dada una matriz  , se define el e-pseudoespectro de , denotándolo como <= /span> , de la siguiente man= era:

<= /span>

Los diferentes métod= os para el cálculo del pseudoespectro de una matriz usan estas definiciones u otras equivalentes. En este trabajo se propone un esquema para el cálculo = del pseudoespectro usando esta última definición basada en el valor singular = mínimo de la matriz .

Representación Gráfica del pseudoespectro<= /b>

Uno de los aportes del pseudoespectro es que el análisis puede realizarse al observar la gráfica que se genera variando los valores de é= psilon ( ). Cada valor de  representa una perturbación de la matr= iz  que genera una curva en el plano comple= jo que encierra el área donde se encontrarían los autovalores de la matriz pertu= rbada. La curva representa entonces la frontera de movimiento de los autovalores e= n el plano complejo.

Si una matriz u operador  es normal, es decir, si posee una base ortogonal de autovectores, entonces su pseudoespectro (en norma 2), está conformado por bolas cerradas de radio  alrededor de los autovalores. Para el c= aso de matrices u operadores no normales, el pseudoespectro se presenta de forma m= enos predecible, lo cual hace su análisis más interesante.

La figura 1 muestra curvas de aproximación del pseudoespectro= de una matriz hermitiana de orden  conocida como matriz ‘smoke’ (a), y una matriz Toepliz no simétrica de orde= n  conocida como ‘grcar’ (b). Ambas ma= trices son usadas frecuentemente en pruebas de algoritmos para el cálculo del pseudoespectro debido a su carácter normal y no-normal que se refleja en l= as curvas de su pseudoespectro. Información más detallada de las caracterís= ticas de estas matrices, y las aplicaciones que modelan, se sugiere ver la págin= a de Pseudospectra Gateway, mantenida por Embree & Trefethen (2021).

=

Figura 1. Pseudoespectro de matrices Normales (a) y No-Normales (b)

Fuente:= Elaboración propia.

Las gráficas se presentan en el plano complejo, y los puntos negros representa= n los autovalores de la matriz, mientras que los trazos coloreados representan los límites del pseudoespectro para cada valor de épsilon, el cual se indica = en la barra de colores a la derecha. Por ejemplo, la curva más externa de color amarillo representa una perturbación de la matriz en = , mientras que la cur= va en color azul representa una perturbación en = . Se puede apreciar q= ue mientras más normal[5] sea la matriz, las cur= vas más cercanas a los autovalores son aquellas correspondientes a valores de =  y a partir de allí siguen la trayector= ia definida por una curva en el plano complejo para la cual el valor = σmin(zI - A)  sea constante, ver por ejemplo Lui (1997).  La principal debilidad de estas implementaciones es la obtención del punto en la frontera del pseudoespect= ro. Adicionalmente, el tiempo de cómputo se incrementa cuando el pseudoespectro tiene componentes conexas, ya que cada componente debe ser tratada particularmente.

Por otra parte, tenemos los algoritmos de malla, que tienen un esquema simple y organizado de ejecución. En estos se define una región de interés en el plano complejo, la cual se discretiza mediante una malla tan refinada como se desee y por cada punto de la malla se calcula . La elección de la región del plano complejo y la definición= del número de puntos a procesar son claves en estos métodos, a fin de obtener contornos con la información deseada.

A continuación, en la figura 3, adaptada de Otero et. al (201= 5), se presenta un esquema básico del cálculo, basados en algoritmo de malla,= para el cálculo del pseudoespectro. La idea general de estos esquemas fue propu= esta por Trefethen (1999). Algunos autores han implementado con éxito este tipo= de métodos, véase por ejemplo Trefethen & Wright (2001) y las referencia= s en Trefethen & Embree (2005).

Figura = 3. Algorit= mo propuesto para matrices de gran tamaño

Fuente: Elaboración propia.

 =

Los algoritmos que proponemos contienen intrínsecamente paralelismo de grano grueso (mínima comunicación entre unidades) una vez = que están basados en la discretización de la región de interés, y paralelis= mo fino (mucha comunicación entre subtareas) en las operaciones matriz-vector que = se utilizan para el cálculo de . En las próximas secciones se presenta el método de proyección que se propon= e para obtener matrices de menor dimensión que contengan información espectral d= e la matriz original A, y la propuesta de discretización de la región de inter= és K.

Proyecciones en espacios de Krylov

Los métodos basados en Arnoldi se fundamentan en la descomposición Hessenberg= de la matriz A o , donde V es u= na matriz Ortogonal; es decir,  y  H  es una matriz Hessenberg superior. Esta descomposición es ideal para la resolución simultánea de los problemas <= /span>  y = , ya que provee un mecanismo mediante el cual se trata un sistema de menor dimensión, además= de tomar ventaja de realizar operaciones del tipo producto matriz-vector. Para= una mejor comprensión de esta descomposición se sugiere ver Treffethen & = Bau (1997) y Golub & Van Loan (1996).

Pa= ra obtener la descomposición de Hessenberg , el método de Arnol= di construye V usando una base ortogonal del subespacio de Krylov:=

En este caso, dado un vector inicial se añade un nuevo ve= ctor a la base en cada iteración, multiplicando el último vector por la matriz= A. En la m-ésima iteración de este algoritmo se satisface la ecuación =  llamada factorización de Arnoldi de orden m, donde  es una matriz de orden ,  y    es una ma= triz Hessenberg de orden m.

Cuando la matriz  es simétrica, la matriz  toma la forma de una tridiagonal y se h= abla entonces del método de Lanczos. En el caso que nos compete se recomienda usar el mÃ= ©todo de Lanczos inverso para hallar el valor singular mínimo de   o de su proyección , considerando que es= te cálculo se traduce en hallar los autovalores de la matriz   o de , las cuales son simétricas.

La precisión y la velocidad de convergencia de este tipo de métodos mejora con el incremento de . Sin embargo, cuando=  aumenta, también aumenta el número de= vectores de Arnoldi y el tamaño de la matriz Hessenberg. Esto compromete los recurs= os de memoria y el costo de CPU, por lo que en la práctica se sugiere mantener <= /span>  en un valor bajo  hasta lograr una precisión aceptable, = o en algunos casos se produce un reinicio del proceso usando como vector inicial= la última información obtenida. En la figura 4 se muestra el tiempo de cómp= uto de la factorización a medida que se incrementa el valor de  para una matriz aleatoria de orden . Se usa el método de Arnoldi con reinicio implícito propuesto por Sorensen (1997), y en todas l= as pruebas el cómputo se detiene al alcanzar una tolerancia de . Implementaciones en bloque para matrices no simétricas y de gran magnitud también han sido propuestas, ver por el ejemplo el trabajo de Castillo (2004).

Figura 4. Tiempo de CPU: Iteración m de Arnoldi

Fuente: Elaboración propia.

 

Discretización de la región de interés

De acuerdo a la aplicación, el investigador pudiera estar interesado en la pa= rte del espectro donde los autovalores toman un valor en particular, por ejempl= o, los de mayor módulo en estudios de estabilidad, o para el análisis de convergencia de métodos iterativos. Este interés define la región en el = plano en la cual se analiza el pseudoespectro.

Una vez definida la región de interés  el próximo paso será la discretizaciÃ= ³n o definición de una malla de puntos. Supongamos que se define una malla de dimensión en la región de interés , tal como se ilustra= en la figura 5(a), adaptada de Guevara (2012).

Cada punto  con ,  de esta malla, representa a un número<= span style=3D'mso-spacerun:yes'>  con a  

Figura 5(a).  Malla de puntos en la región

Fuente: Elaboración prop= ia.

 

 

Consideremos también que tenemos una arquitectura paralela co= n  procesadores, etiquetados , , …,  subregiones en , etiquetadas desde <= /span>  hasta , tal como se presenta en la figura 5(b), adaptada de Guevara = (2012).

Figura 5(b). Distribución de la carga en los procesadores

Fuente: Elaboración propia.

Cada subregión se asocia a un proceso y cada proceso calcula =  para todos los puntos  de su subregión. Este cálculo es simu= ltáneo y representa un paralelismo de grano grueso soportado por la independencia de= los datos. Los procesos terminan al enviar los valores singulares calculados al proceso maestro que asumirá la generación del gráfico del pseudoespectro= . Este esquema, basado en la discretización mostrada en la figura 5, permite dism= inuir significativamente el tiempo de cómputo.

Siempre podremos hacer una distribución equitativa del númer= o de puntos que atiende cada procesador si el número total de puntos mx . ny   es divisible por . En caso de que de <= /span>   no sea divisible por  podríamos por ejemplo recargar al proc= esador  con  puntos adicionales. <= /p>

Ad= icionalmente, cada cálculo de  se hace con un código basado en multiplicaciones matriz-vector, que se propone hacer en paralelo también representando un nivel de paralelismo de grano fino.

Unidades de Procesamiento Gráfico (GPUs)

En= la actualidad los computadores personales y de oficina poseen múltiples núcl= eos o cores (procesadores) en la llamada Unidad Central de Procesamiento o CPU, además la tecnología incorpora, a bajo costo, tarjetas gráficas con una = Unidad de Procesamiento Gráfico o GPU que también contiene muchos núcleos diseÃ= ±ados para el cómputo masivo, por lo que pudiéramos aceptar que la computación paralela está a nuestro alcance.

Los procesadores que se encuentran en la GPU pueden ser administrados eficientemente desde la unidad central de procesamiento o CPU para realizar cómputo intensivo. La idea es sustituir los procesos centralizados por procesos distribuidos entre los procesadores (cores) del = CPU y los de las GPU. Para este fin se utilizan arquitecturas vectoriales o paralelas manejadas por plataformas de enlace que permiten consolidar resultados entre CPU y GPU.

En el año 2006 la compañía NVIDIA introduce la Arquitectura= de Dispositivos de Cómputo Unificado o CUDA (por sus siglas en inglés), y ha= ce posible el manejo efectivo de los núcleos de la GPU a través de un modelo= de programación que permite la interacción CPU-GPU desde un programa de usua= rio. Detalles de esta tecnología, su uso y programación pueden encontrarse en = la guía de programación de CUDA en NVIDIA (2021).

Con estas herramientas tecnológicas el potencial de la tarjeta gráfica se extiende a la implementación de algoritmos altamente paraleliz= ables que consumen mucho tiempo de CPU; tal es el caso de los algoritmos basados = en productos de matrices y de vectores, que aparecen frecuentemente en muchas aplicaciones del álgebra lineal, como el cálculo del espectro y/o el pseudoespectro de matrices dispersas y de gran magnitud.<= /p>

Algunos investigadores se han dedicado desde entonces a la implementación de estos algoritmos, véase por ejemplo el trabajo de Angel= es et al. (2011).  Adicionalmente existen librerías de acceso libre con los códigos más usados. Estas librerías p= oseen interfaces para brindar flexibilidad en la programación y permitir la ejec= ución de instrucciones para la CPU y para la GPU en el mismo programa. Como ejemp= lo, véase Thrust, librería desarrollada por Bell y Hoberock (2011), que permite manejar estructuras de datos en algoritmos en paralelo, y la librería Cusp que contiene módulos para el manejo eficient= e de cálculo matricial y la ejecución de métodos clásicos para la resolució= n de sistemas lineales, desarrollada por Bell y Garland (2009).

El modelo de arquitectura induce a dejar las tareas con cálcu= los secuenciales, como el manejo de entrada y salida de datos, para que se ejec= uten en la CPU (usualmente llamada host) y las tareas de cómputo masivo paralel= izable que se ejecuten sobre la GPU (denominada device). Actualmente se cuenta con computadores de escritorio de 8 cores con una tarjeta Nvidia de miles de co= res y este gap seguirá ampliándose en el futuro.

Cada núcleo de la GPU tiene recursos compartidos, como regist= ros y memoria, integrados en el mismo chip, lo cual permite que las tareas (hilos) que se ejecutan en un mismo núcleo compartan datos sin usar un bus de memo= ria. La arquitectura permite al programador usar funciones escritas en lenguajes clásicos de alto nivel como C y C++ (conocidas como kernels) que luego se ejecutan en paralelo en la GPU como un conjunto de hilos que organizan dent= ro una jerarquía de bloques.

Para la programación en CUDA se debe considerar los siguientes elementos: jerarquía de bloques de hilos, compartición de memoria y sincronización.

Estos elementos dirigen la programación hacia la división del problema en tareas que pueden ser ejecutadas en paralelo por bloques de hil= os independientes, que a su vez podrían agruparse en una malla de procesadore= s (grid), véase figura 6. La sincronización es manejad= a por el programador usando herramientas de CUDA que garantizan la escalabilidad; es decir, la aplicación de usuario debe soportar el incremento de cores en la= GPU sin necesidad de reprogramación.

Figura 6.  Malla de bloques de hilos

Fuente:= Adaptada de NVIDIA (2021).

La propuesta de llevar estos cálculos a un esquema híbrido q= ue involucre el cómputo masivo en la GPU no solo se sustenta en la posibilida= d de incrementar el nivel de paralelismo y disminuir los tiempos de respuesta, sino que adem= ás el poder de cómputo de los cores en las actuales GPU es muy superior a los= de la CPU, véase por ejemplo el trabajo de Arce et. al (2011).

Resultados

Una vez analizados cada componente involucrado en el cálculo = del pseudoespectro, se resume a continuación las bases de la propuesta:

1)   Entrada de datos (CPU):

1.1) Lectura o definición de la matriz .

1.2)  Lectura o definición de los parámetros de Arnoldi.

1.3) Distribución de la matriz  en= los cores. (CPU-GPU).

2)  Proyección de la matriz , de orden , en un espacio de dimensión =  (CPU-GPU).

2.1) Distribución de la matriz  en= los cores. (CPU-GPU).

2.2)  Ejecución de bloques de operaciones matriz-vector para aplicación del método de Arnoldi con reinicio implíci= to a la matriz . (GPU).

2.3)  Generación de matriz    y = su distribución o carga en los cores de la GPU. (CPU-GPU)

3)   Discretización o definición de malla.=

3.1)  Lectura o definición de parámetros de la m= alla. (CPU).

3.2)  Distribución de regiones en los cores la GP= U. (GPU)

4)   Cálculo del mínimo valor singular de los p= untos en cada región (GPU).

3. <= /span> <= /span>

4. <= /span> <= /span>

4.1) Para cada punto  de= la subregión, ejecutar el método de Lanczos para hallar el valor singular mÃ= ­nimo de  ,<= span lang=3DES style=3D'font-size:12.0pt;font-family:"Times New Roman",serif;mso= -fareast-font-family: "Times New Roman";color:black'> a través del autovalor máximo de <= /span> .<= span lang=3DES style=3D'font-size:12.0pt;font-family:"Times New Roman",serif;mso= -fareast-font-family: "Times New Roman";color:black'>

4.2) Envío de la submatriz correspondiente a los=    de= cada  en= la subregión.

5)   Visualización de contornos (CPU)=

5.1)  Graficación de curvas de contornos <= /span>  co= n .

Conclusiones

·      =    Se han descrito los aspectos fundamentales de una propuesta pa= ra el cálculo del pseudoespectro de matrices usando un esquema híbrido CPU-G= PU que añade un nivel adicional de paralelismo con el fin de acelerar este cálcu= lo que generalmente resulta costoso computacionalmente. Adicionalmente se han expuestos métodos que han logrado tener éxito en todas las implementacion= es paralelas del pseudoespectro en máquinas de alto rendimiento. Se espera implementar estos algoritmos en un futuro inmediato bajo una plataforma CUDA con MPI usando GPU dispuestas en tarjetas NVIDIA en computadores de gama me= dia o baja, a fin de probar el potencial de la propuesta.

·=       =    La propuesta conserva características generales y cada módulo puede ser implementado o bien utilizando los métodos recomendados en cada tarea, u otros métodos de preferencia.

 

El presente documento estuvo dirigido a:

1.   Establecer los fundamentos teóricos para el cálculo del pseudoespectro; cubriendo detalles sobre su = uso e importancia.

2.   Justificar el uso de la herramienta y de los métodos de cálculo.

3.   Desarrollar una propuesta general que contemple las distintas etapas del cálculo del pseudoespectro.=

4.   Proponer algoritmos bási= cos en cada actividad.

5.   Describir las caracterís= ticas de las unidades de procesamiento gráfico, y su utilidad para hacer cálcul= os en paralelo.

6.   Señalar detalles de implementación y consideraciones de rigor en base a la arquitectura propue= sta.

 

Referencias bibliográficas

Ãngeles, M., Flores, G., Vidal,= A. (2011).  Implementación CPU-GPU y comparativa de las bibliotecas BLAS-CUBLAS, LAPACK-CULA, Universidad Politécnica de Valencia.

Bekas, C., Kokiopoulou, E., Kouti= s, I., Gallopoulus, E. (2001) Towards the effective Parallel Computation of Matrix Pseudospectra. IC’s 01: Proceeding of= the 15th International Conference on Supercomputing, June 2001, 203-226.

Bekas, C., Kokiopoulou, E., Gallo= poulos E., Simoncini, V., Parallel Computation of Pseudospectra using Transfer Functions on a MATLAB-MPI Cluster Platform. Lecture Notes in Computer Science. DOI:10.1007/3-540-45825-5_35.

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Otero, B., Astudillo, R., Castillo, Z. (2015). Un esquema paralelo para el cálculo del pseudoespectro de matrices de gran magnitud. Revista Internacional de Métodos Numéricos = para el Cálculo y Diseño en Ingeniería. 31-1, 8-12.<= o:p>

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PARA CITAR EL ARTÃCULO INDEXADO.

 

 

Castillo Marrero, Z. N., Colmenares Pacheco, G. A., Valverde Aguirre= , P. E., & Cevallos Vique, V. O. (2021). Una pro= puesta para el cálculo del pseudoespectro en unidades= de procesamiento gráfico. ConcienciaDigital, 4(2.= 1), 6-20. ht= tps://doi.org/10.33262/concienciadigital.v4i2.1.1703<= /p>

 

 


 

 

 

El artículo que se publica es de exclusiva responsabilidad de los autores y no necesariamente reflejan el pensamiento de la Revi= sta Conciencia Digital.

 

El artículo queda en propiedad de la revista y, por tanto, su publicación pa= rcial y/o total en otro medio tiene que ser autorizado por el director de la Revista Conciencia Digital.<= /o:p>

 

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[1] Escuela Superior Politécnica de Chimborazo, Facultad de Ciencias, Escuela de Estadística, Grupo CIDED. Riobamba, Ecuador, zenaida.castillo= @espoch.edu.ec, = https://orcid.org/0000-0002-4424-8652<= /span>

[2] Universidad de Investigación Experimental Yachay, Escuela de Matemáticas y Ciencias Computacionales, Urcuquí, Ecuador, gcolmenares@yach= aytech.edu.ec, https://orcid.org= /0000-0003-4789-0859

[3] Escuela Superior Politécnica de Chimborazo, Facultad de Ciencias, Grupo CIDED. Riobamba, Ecuador, paulina.valverde@espoch.edu.ec<= span lang=3DES style=3D'font-family:"Times New Roman",serif'>, https://orcid.org= /0000-0003-0458-7083

[4] Escuela Superior Politécnica de Chimborazo, Facultad de Administración de Empresas, Escuel= a de Finanzas, Grupo CIDED, Riobamba, Ecuador, victor.cevallos@= espoch.edu.ec, https://orcid.org= /0000-0001-5525-5818

[5] La n= orma de (ATA – AAT) proporciona una medida de la no-normalidad.

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