System, method and computer program product for dense/sparse linear system solver accelerator

A computer program product for linear system solving that interacts with a cloud computing environment server, the computer program product comprising a computer-readable storage medium having program instructions embodied therewith, the program instructions executable by a computer to cause the computer to perform:

• calculating, via a cloud on-demand self-service that communicates with the cloud computing environment server, a matrix factorization for a matrix in a pair of matrices, in a form of a lower, a diagonal, and an upper (LDU) decomposition by interacting directly with a system memory through a Coherent Accelerator Processor Interface (CAPI) interface;
• first solving, via the cloud on-demand self-service that communicates with the cloud computing environment server, a first expression for a first value calculated from using the first expression using a substitution module based on the LDU decomposition to create a first result;
• dividing, via the cloud on-demand self-service that communicates with the cloud computing environment server, the first result by values stored in the diagonal of the matrix to obtain a second result;
• second solving, via the cloud on-demand self-service that communicates with the cloud computing environment server, a second expression for a second value calculated from using the second expression where a processing of the diagonal is skipped by using the second result;
• and
• determining, via the cloud on-demand self-service that communicates with the cloud computing environment server, a solution set for the pair of matrices in the linear system using the second expression,
• wherein the calculating proceeds to calculate a second matrix factorization for a second matrix while the first solving, the dividing, and the second solving are working with the matrix on the system memory, and
• wherein the first solving and the second solving are performed using a same substitution module,
• further comprising streaming the solution from the cloud to a device,
• wherein the cloud computing environment server comprises a cloud computing model of a service delivery comprising two or more clouds of a private cloud, a community cloud, and a public cloud that remain unique entities but are bound together by technology that enables data and application portability that results in load-balancing between the two or more clouds.

The computer program product claim of 10, wherein, in the first solving, the first expression is Ly=b and is solved for the first value of b using the substitution module to create the result, and

• wherein the substitution module performs a forward substitution using a same module as used in the LDU decomposition to calculate a third expression of
• b
• i
• -
• ∑
• k
• =
• 0
• i
• =
• 1
• ⁢
• ⁢
• L
• i
• ⁢
• k
• ⁢
• x
• k
• .

The computer program product claim of 12, wherein the second solving solves the second expression of

• Ux
• =
• y
• D
• for the second value of y, and
• wherein, in a forward transformation (FTRAN) in the second solving, “y1=b1” is set as “L11=1”, then “y2=b2−y1L21” and continuing for each element of y using
• y
• i
• =
• b
• i
• -
• ∑
• k
• =
• 0
• i
• =
• 1
• ⁢
• L
• ik
• ⁢
• y
• k
• until a last element is reached.

A linear system solving system, said system comprising:

• a processor;
• and
• a memory, the memory storing instructions to cause the processor to perform:
• calculating, via a cloud on-demand self-service that communicates with the cloud computing environment server, a matrix factorization for a matrix in a pair of matrices, in a form of a lower, a diagonal, and an upper (LDU) decomposition by interacting directly with a system memory through a Coherent Accelerator Processor Interface (CAPI) interface;
• first solving, via the cloud on-demand self-service that communicates with the cloud computing environment server, a first expression for a first value calculated from using the first expression using a substitution module based on the LDU decomposition to create a first result;
• dividing, via the cloud on-demand self-service that communicates with the cloud computing environment server, the first result by values stored in the diagonal of the matrix to obtain a second result;
• second solving, via the cloud on-demand self-service that communicates with the cloud computing environment server, a second expression for a second value calculated from using the second expression where a processing of the diagonal is skipped by using the second result;
• and
• determining, via the cloud on-demand self-service that communicates with the cloud computing environment server, a solution set for the pair of matrices in the linear system using the second expression,
• wherein the calculating proceeds to calculate a second matrix factorization for a second matrix while the first solving, the dividing, and the second solving are working with the matrix on the system memory, and
• wherein the first solving and the second solving are performed using a same substitution module,
• further comprising streaming the solution from the cloud to a device,
• wherein the cloud computing environment server comprises a cloud computing model of a service delivery comprising two or more clouds of a private cloud, a community cloud, and a public cloud that remain unique entities but are bound together by technology that enables data and application portability that results in load-balancing between the two or more clouds.