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Overview

The interface for MAGMA is similar to LAPACK, to facilitate porting existing codes.

Many routines have the same base names and the same arguments as LAPACK. In some cases, MAGMA needs larger workspaces or some additional arguments in order to implement an efficient algorithm.

There are several classes of routines in MAGMA:

1. Driver routines – Solve an entire problem.
2. Computational routines – Solve one piece of a problem.
3. BLAS routines – Basic Linear Algebra Subroutines. These form the basis for linear algebra algorithms.
4. Auxiliary routines – Additional BLAS-like routines, many originally defined in LAPACK.
5. Utility routines – Additional routines, many specific to GPU programming.

A brief summary of routines is given here. Full descriptions of individual routines are given in the Modules section.

To start, in C or C++, include the magma_v2.h header.

    #include <magma_v2.h>

Driver & computational routines have a magma_ prefix. These are generally hybrid CPU/GPU algorithms. A suffix indicates in what memory the matrix starts and ends, not where the computation is done.

Suffix | Example | Description
--- | --- | ---
none | magma_dgetrf | hybrid CPU/GPU routine where the matrix is initially in CPU host memory.
_m | magma_dgetrf_m | hybrid CPU/multiple-GPU routine where the matrix is initially in CPU host memory.
_gpu | magma_dgetrf_gpu | hybrid CPU/GPU routine where the matrix is initially in GPU device memory.
_mgpu | magma_dgetrf_mgpu | hybrid CPU/multiple-GPU routine where the matrix is distributed across multiple GPUs' device memories.

In general, MAGMA follows LAPACK's naming conventions. The base name of each routine has a one letter precision (occasionally two letters), two letter matrix type, and usually a 2-3 letter routine name. For example, DGETRF is D (double-precision), GE (general matrix), TRF (triangular factorization).

Precision | Description
--- | ---
s | single real precision (float)
d | double real precision (double)
c | single-complex precision (magmaFloatComplex)
z | double-complex precision (magmaDoubleComplex)
sc | single-complex input with single precision result (e.g., scnrm2)
dz | double-complex input with double precision result (e.g., dznrm2)
ds | mixed-precision algorithm (double and single, e.g., dsgesv)
zc | mixed-precision algorithm (double-complex and single-complex, e.g., zcgesv)

Matrix type | Description
--- | ---
ge | general matrix
sy | symmetric matrix, can be real or complex
he | Hermitian (complex) matrix
po | positive definite, symmetric (real) or Hermitian (complex) matrix
tr | triangular matrix
or | orthogonal (real) matrix
un | unitary (complex) matrix

# Driver routines

Driver routines solve an entire problem.

Name | Description
--- | ---
gesv | solve linear system, AX = B, A is general (non-symmetric)
posv | solve linear system, AX = B, A is symmetric/Hermitian positive definite
hesv | solve linear system, AX = B, A is symmetric indefinite
gels | least squares solve, AX = B, A is rectangular
geev | non-symmetric eigenvalue solver, AX = X Lambda
syev/heev | symmetric eigenvalue solver, AX = X Lambda
syevd/heevd | symmetric eigenvalue solver, AX = X Lambda, using divide & conquer
sygvd/hegvd | symmetric generalized eigenvalue solver, AX = BX Lambda
gesvd | singular value decomposition (SVD), A = U Sigma V^H
gesdd | singular value decomposition (SVD), A = U Sigma V^H, using divide & conquer

# Computational routines

Computational routines solve one piece of a problem. Typically, driver routines call several computational routines to solve the entire problem. Here, curly braces { } group similar routines. Starred * routines are not yet implemented in MAGMA.

| Name | Description
--- | --- | ---
Triangular factorizations | getrf, potrf, hetrf | triangular factorization (LU, Cholesky, Indefinite)
getrs, potrs, hetrs | triangular forward and back solve
getri, potri | triangular inverse
getf2, potf2 | triangular panel factorization (BLAS-2)
Orthogonal factorizations | ge{qrf, qlf, lqf, rqf*} | QR, QL, LQ, RQ factorization
geqp3 | QR with column pivoting (BLAS-3)
or{mqr, mql, mlq, mrq*} | multiply by Q after factorization (real)
un{mqr, mql, mlq, mrq*} | multiply by Q after factorization (complex)
or{gqr, gql*, glq*, grq*} | generate Q after factorization (real)
un{gqr, gql*, glq*, grq*} | generate Q after factorization (complex)
Eigenvalue & SVD | gehrd | Hessenberg reduction (in geev)
sytrd/hetrd | tridiagonal reduction (in syev, heev)
gebrd | bidiagonal reduction (in gesvd)

There are many other computational routines that are mostly internal to MAGMA and LAPACK, and not commonly called by end users.

# BLAS routines

BLAS routines follow a similar naming scheme: precision, matrix type (for level 2 & 3), routine name. For BLAS routines, the magma_ prefix indicates a wrapper around CUBLAS (e.g., magma_zgemm calls cublasZgemm), while the magmablas_ prefix indicates our own MAGMA implementation (e.g., magmablas_zgemm). All MAGMA BLAS routines are GPU native and take the matrix in GPU memory. The descriptions here are simplified, omitting scalars (alpha & beta) and transposes.

## BLAS-1: vector operations

These do O(n) operations on O(n) data and are memory-bound.

Name | Description
--- | ---
copy | copy vector, y = x
scal | scale vector, y = alpha*y
swap | swap two vectors, y <---> x
axpy | y = alpha*x + y
nrm2 | vector 2-norm
amax | vector max-norm
asum | vector one-norm
dot | dot product (real), x^T y
dotu | dot product (complex), unconjugated, x^T y
dotc | dot product (complex), conjugated, x^H y

## BLAS-2: matrix-vector operations

These do O(n^2) operations on O(n^2) data and are memory-bound.

Name | Description
--- | ---
gemv | general matrix-vector product, y = A*x
symv, hemv | symmetric/Hermitian matrix-vector product, y = A*x
syr, her | symmetric/Hermitian rank-1 update, A = A + x*x^H
syr2, her2 | symmetric/Hermitian rank-2 update, A = A + x*y^H + y*x^H
trmv | triangular matrix-vector product, y = A*x
trsv | triangular solve, one right-hand side (RHS), solve Ax = b

## BLAS-3: matrix-matrix operations

These do O(n^3) operations on O(n^2) data and are compute-bound. Level 3 BLAS are significantly more efficient than the memory-bound level 1 and level 2 BLAS.

Name | Description
--- | ---
gemm | general matrix-matrix multiply, C = C + A*B
symm, hemm | symmetric/Hermitian matrix-matrix multiply, C = C + A*B, A is symmetric
syrk, herk | symmetric/Hermitian rank-k update, C = C + A*A^H, C is symmetric
syr2k, her2k | symmetric/Hermitian rank-2k update, C = C + A*B^H + B*A^H, C is symmetric
trmm | triangular matrix-matrix multiply, B = A*B or B*A, A is triangular
trsm | triangular solve, multiple RHS, solve A*X = B or X*A = B, A is triangular

# Auxiliary routines

Additional BLAS-like routines, many originally defined in LAPACK. These follow a similar naming scheme: precision, then "la", then the routine name. MAGMA implements these common ones on the GPU, plus adds a few such as symmetrize and transpose.

For auxiliary routines, the magmablas_ prefix indicates our own MAGMA implementation (e.g., magmablas_zlaswp). All MAGMA auxiliary routines are GPU native and take the matrix in GPU memory.

Name | Description
--- | ---
geadd | add general matrices (like axpy), B = alpha*A + B
laswp | swap rows (in getrf)
laset | set matrix to constant
lacpy | copy matrix
lascl | scale matrix
lange | norm, general matrix
lansy/lanhe | norm, symmetric/Hermitian matrix
lantr | norm, triangular matrix
lag2 | convert general matrix from one precision to another (e.g., dlag2s is double to single)
lat2 | convert triangular matrix from one precision to another
larf | apply Householder elementary reflector
larfg | generate Householder elementary reflector
larfb | apply block Householder elementary reflector
larft | form T for block Householder elementary reflector
symmetrize | copy lower triangle to upper triangle, or vice-versa
transpose | transpose matrix

# Utility routines

## Memory Allocation

MAGMA can use regular CPU memory allocated with malloc or new, but it may achieve better performance using aligned and, especially, pinned memory. There are typed versions of these (e.g., magma_zmalloc) that avoid the need to cast and use sizeof, and un-typed versions (e.g., magma_malloc) that are more flexible but require a (void**) cast and multiplying the number of elements by sizeof.

Name | Description
--- | ---
magma_malloc_cpu() | allocate CPU memory that is aligned for better performance & reproducibility
magma_malloc_pinned() | allocate CPU memory that is pinned (page-locked)
magma_malloc() | allocate GPU memory
magma_free_cpu() | free CPU memory allocated with magma_malloc_cpu
magma_free_pinned() | free CPU memory allocated with magma_malloc_pinned
magma_free() | free GPU memory allocated with magma_malloc

where * is one of the four precisions, s d c z, or i for magma_int_t, or none for an un-typed version.

## Communication

The name of communication routines is from the CPU's point of view.

Name | Description
--- | ---
setmatrix | send matrix to GPU
setvector | send vector to GPU
getmatrix | get matrix from GPU
getvector | get vector from GPU

# Data types & complex numbers

## Integers

MAGMA uses magma_int_t for integers. Normally, this is mapped to the C/C++ int type. Most systems today use the LP64 convention, meaning long and pointers are 64-bit, while int is 32-bit.

MAGMA also supports the ILP64 convention as an alternative, where int, long, and pointers are all 64-bit. To use this, we typedef magma_int_t to be long long. To use ILP64, define MAGMA_ILP64 or MKL_ILP64 when compiling, and link with an ILP64 BLAS and LAPACK library; see make.inc.mkl-ilp64 for an example.

## Complex numbers

MAGMA supports complex numbers. Unfortunately, there is not a single standard for how to implement complex numbers in C/C++. Fortunately, most implementations are identical on a binary level, so you can freely cast from one to another. The MAGMA types are: magmaFloatComplex, which in CUDA MAGMA is a typedef of cuFloatComplex, and magmaDoubleComplex, which in CUDA MAGMA is a typedef of cuDoubleComplex.

For C, we provide macros to manipulate complex numbers. For C++ support, include the magma_operators.h header, which provides overloaded C++ operators and functions.

C macro | C++ operator | Description
--- | --- | ---
c = MAGMA_*_MAKE(r,i) |  | create complex number from real & imaginary parts
r = MAGMA_*_REAL(a) | r = real(a) | return real part
i = MAGMA_*_IMAG(a) | i = imag(a) | return imaginary part
c = MAGMA_*_NEGATE(a) | c = -a; | negate
c = MAGMA_*_ADD(a,b) | c = a + b; | add
c = MAGMA_*_SUB(a,b) | c = a - b; | subtract
c = MAGMA_*_MUL(a,b) | c = a * b; | multiply
c = MAGMA_*_DIV(a,b) | c = a / b; | divide
c = MAGMA_*_CNJG(a) | c = conj(a) | conjugate
r = MAGMA_*_ABS(a) | r = fabs(a) | 2-norm, sqrt( real(a)^2 + imag(a)^2 )
r = MAGMA_*_ABS1(a) | r = abs1(a) | 1-norm, abs(real(a)) + abs(imag(a))
. Constants |  | Description
c = MAGMA_*_ZERO |  | zero
c = MAGMA_*_ONE |  | one
c = MAGMA_*_NAN |  | not-a-number (e.g., 0/0)
c = MAGMA_*_INF |  | infinity (e.g., 1/0, overflow)

where * is one of the four precisions, S D C Z.

# Conventions for variables

Here are general guidelines for variable names; there are of course exceptions to these.

- Uppercase letters indicate matrices: A, B, C, X.
- Lowercase letters indicate vectors: b, x, y, z.
- "d" prefix indicates matrix or vector on GPU device: dA, dB, dC, dX; db, dx, dy, dz.
- Greek words indicate scalars: alpha, beta.
- m, n, k are matrix dimensions.

Typically, the order of arguments is:

- options (uplo, etc.)
- matrix sizes (m, n, k, etc.),
- input matrices & vectors (A, lda, x, incx, etc.)
- output matrices & vectors
- workspaces (work, lwork, etc.)
- info error code

LAPACK and MAGMA use column-major matrices. For matrix X with dimension (lda,n), element X(i, j) is X[ i + j*lda ]. For symmetric, Hermitian, and triangular matrices, only the lower or upper triangle is accessed, as specified by the uplo argument; the other triangle is ignored.

lda is the leading dimension of matrix A; similarly ldb for B, ldda for dA, etc. It should immediately follow the matrix pointer in the argument list. The leading dimension can be the number of rows, or if A is a sub-matrix of a larger parent matrix, lda is the leading dimension (e.g., rows) of the parent matrix.

On the GPU, it is often beneficial to round the leading dimension up to a multiple of 32, to provide better performance. This aligns memory reads so they are coalesced. This is provided by the magma_roundup function:

    ldda = magma_roundup( m, 32 );

The formula ((m + 31)/32)*32 also works, relying on floored integer division, but the roundup function is clearer to use.

On the CPU, it is often beneficial to ensure that the leading dimension is not a multiple of the page size (often 4 KiB), to minimize TLB misses.

For vectors, incx is the increment or stride between elements of vector x. In all cases, incx != 0. In most cases, if incx < 0, then the vector is indexed in reverse order, for instance, using Matlab notation,

    incx =  1   means   x( 1 : 1 : n     )
    incx =  2   means   x( 1 : 2 : 2*n-1 )

while

    incx = -1   means   x( n     : -1 : 1 )
    incx = -2   means   x( 2*n-1 : -2 : 1 )

For several routines (amax, amin, asum, nrm2, scal), the order is irrelevant, so negative incx are not allowed; incx > 0.

# Constants

MAGMA defines a few constant parameters, such as MagmaTrans, MagmaNoTrans, that are equivalent of CBLAS and LAPACK parameters. The naming and numbering of these parameters follow that of CBLAS from Netlib and the C Interface to LAPACK from Netlib, and PLASMA.

MAGMA includes functions, lapack_xyz_const(), which take MAGMA's integer constants and return LAPACK's string constants, where xyz is a MAGMA type such as uplo, trans, etc. From the standpoint of LAPACK, only the first letter of each string is significant. Nevertheless, the functions return meaningful strings, such as "No transpose", "Transpose", "Upper", "Lower", etc. Similarly, there are functions to go from MAGMA's integer constants to CBLAS, OpenCL's clBLAS, and CUDA's cuBLAS integer constants.

There are also functions, magma_xyz_const(), to go in the opposite direction, from LAPACK's string constants to MAGMA's integer constants.

The most common constants are those defined for BLAS routines:

- enum { MagmaNoTrans, MagmaTrans, MagmaConjTrans } magma_order_t

  Whether a matrix is not transposed, transposed, or conjugate-transposed. For a real matrix, Trans and ConjTrans have the same meaning.

- enum { MagmaLower, MagmaUpper, MagmaFull } magma_uplo_t

  Whether the lower or upper triangle of a matrix is given, or the full matrix.

- enum { MagmaLeft, MagmaRight } magma_side_t

  Whether the matrix is on the left or right.

- enum { MagmaUnit, MagmaNonUnit } magma_diag_t

  Whether the diagonal is assumed to be unit (all ones) or not.

Additional constants for specific routines are defined in the documentation for the routines.

Because MAGMA, CBLAS, LAPACK, CUBLAS, and clBlas use potentially different constants, converters between them are provided.

These convert LAPACK constants to MAGMA constants. Note that the meaning of LAPACK constants depends on the context: 'N' can mean False, NoTrans, NonUnit, NoVec, etc. Here, curly braces { } group similar constants.

. | Function | . | Description
--- | --- | --- | ---
magma_bool_t | magma_bool_const | ( character ) | Map 'N', 'Y' to MagmaTrue, MagmaFalse
magma_order_t | magma_order_const | ( character ) | Map 'R', 'C' to MagmaRowMajor, MagmaColMajor
magma_trans_t | magma_trans_const | ( character ) | Map 'N', 'T', 'C' to MagmaNoTrans, MagmaTrans, MagmaConjTrans
magma_uplo_t | magma_uplo_const | ( character ) | Map 'L', 'U' to MagmaLower, MagmaUpper
magma_diag_t | magma_diag_const | ( character ) | Map 'N', 'U' to MagmaNonUnit, MagmaUnit
magma_side_t | magma_side_const | ( character ) | Map 'L', 'R' to MagmaLeft, MagmaRight
magma_norm_t | magma_norm_const | ( character ) | Map 'O', '1', '2', 'F', 'E', 'I', 'M' to Magma{One, Two, Frobenius, Inf, Max}Norm
magma_dist_t | magma_dist_const | ( character ) | Map 'U', 'S', 'N' to MagmaDist{Uniform, Symmetric, Normal}
magma_vec_t | magma_vec_const | ( character ) | Map 'V', 'N', 'I', 'A', 'S', 'O' to MagmaVec, Magma{No, I, All, Some, Overwrite}Vec
magma_range_t | magma_range_const | ( character ) | Map 'A', 'V', 'I' to MagmaRange{All, V, I}
magma_vect_t | magma_vect_const | ( character ) | Map 'Q', 'P' to MagmaQ, MagmaP
magma_direct_t | magma_direct_const | ( character ) | Map 'F', 'B' to MagmaForward, MagmaBackward
magma_storev_t | magma_storev_const | ( character ) | Map 'C', 'R' to MagmaColumnwise, MagmaRowwise

These do the inverse map, converting MAGMA to LAPACK constants. From the standpoint of LAPACK, only the first letter of each string is significant. Nevertheless, the functions return meaningful strings, such as "No transpose", "Transpose". Substitute lapacke for lapack to get version that returns single char instead of string (const char*).

. | Function | . | Description
--- | --- | --- | ---
const char* | lapack_bool_const | ( magma_bool_t ) | Inverse of magma_bool_const()
const char* | lapack_order_const | ( magma_order_t ) | Inverse of magma_order_const()
const char* | lapack_trans_const | ( magma_trans_t ) | Inverse of magma_trans_const()
const char* | lapack_uplo_const | ( magma_uplo_t ) | Inverse of magma_uplo_const()
const char* | lapack_diag_const | ( magma_diag_t ) | Inverse of magma_diag_const()
const char* | lapack_side_const | ( magma_side_t ) | Inverse of magma_side_const()
const char* | lapack_norm_const | ( magma_norm_t ) | Inverse of magma_norm_const()
const char* | lapack_dist_const | ( magma_dist_t ) | Inverse of magma_dist_const()
const char* | lapack_vec_const | ( magma_vec_t ) | Inverse of magma_vec_const()
const char* | lapack_range_const | ( magma_range_t ) | Inverse of magma_range_const()
const char* | lapack_vect_const | ( magma_vect_t ) | Inverse of magma_vect_const()
const char* | lapack_direct_const | ( magma_direct_t ) | Inverse of magma_direct_const()
const char* | lapack_storev_const | ( magma_storev_t ) | Inverse of magma_storev_const()
const char* | lapack_const | ( constant ) | Map any MAGMA constant, Magma*, to an LAPACK string constant
char | lapacke_const | ( constant ) | Map any MAGMA constant, Magma*, to an LAPACKE character

To convert MAGMA to Nvidia's CUBLAS constants:

. | Function | . | Description
--- | --- | --- | ---
cublasOperation_t | cublas_trans_const | ( trans ) | Map MagmaNoTrans, MagmaTrans, MagmaConjTrans to CUBLAS_OP_N, CUBLAS_OP_T, CUBLAS_OP_C
cublasFillMode_t | cublas_uplo_const | ( uplo ) | Map MagmaLower, MagmaUpper to CUBLAS_FILL_MODE_LOWER, CUBLAS_FILL_MODE_UPPER
cublasDiagType_t | cublas_diag_const | ( diag ) | Map MagmaNonUnit, MagmaUnit to CUBLAS_DIAG_NON_UNIT, CUBLAS_DIAG_UNIT
cublasSideMode_t | cublas_side_const | ( side ) | Map MagmaLeft, MagmaRight to CUBLAS_SIDE_LEFT, CUBLAS_SIDE_Right

To convert MAGMA to AMD's clBlas constants:

. | Function | . | Description
--- | --- | --- | ---
clblasOrder | clblas_order_const | ( order ) | Map MagmaRowMajor, MagmaColMajor to clAmdBlasRowMajor, clAmdBlasColumnMajor
clblasTranspose | clblas_trans_const | ( trans ) | Map MagmaNoTrans, MagmaTrans, MagmaConjTrans to clAmdBlasNoTrans, clAmdBlasTrans, clAmdBlasConjTrans
clblasUplo | clblas_uplo_const | ( uplo ) | Map MagmaLower, MagmaUpper to clAmdBlasLower, clAmdBlasUpper
clblasDiag | clblas_diag_const | ( diag ) | Map MagmaNonUnit, MagmaUnit to clAmdBlasNonUnit, clAmdBlasUnit
clblasSide | clblas_side_const | ( side ) | Map MagmaLeft, MagmaRight to clAmdBlasLeft, clAmdBlasRight

To convert MAGMA to CBLAS constants:

. | Function | . | Description
--- | --- | --- | ---
enum CBLAS_ORDER | cblas_order_const | ( order ) | Map MagmaRowMajor, MagmaColMajor to CblasRowMajor, CblasColMajor
enum CBLAS_TRANSPOSE | cblas_trans_const | ( trans ) | Map MagmaNoTrans, MagmaTrans, MagmaConjTrans to CblasNoTrans, CblasTrans, CblasConjTrans
enum CBLAS_UPLO | cblas_uplo_const | ( uplo ) | Map MagmaLower, MagmaUpper to CblasLower, CblasUpper
enum CBLAS_DIAG | cblas_diag_const | ( diag ) | Map MagmaNonUnit, MagmaUnit to CblasNonUnit, CblasUnit
enum CBLAS_SIDE | cblas_side_const | ( side ) | Map MagmaLeft, MagmaRight to CblasLeft, CblasRight

# Errors

Driver and computational routines, and a few BLAS/auxiliary routines, currently return errors both as a return value and in the info argument. The return value and info should always be identical. In general, the meaning is as given in this table. Predefined error codes are large negative numbers. Using the symbolic constants below is preferred, but the numeric values can be found in include/magma_types.h.

Info | Description
--- | ---
info = 0 (MAGMA_SUCCESS) | Successful exit
info < 0, but small | For info = -i, the i-th argument had an illegal value
info > 0 | Function-specific error such as singular matrix
MAGMA_ERR_DEVICE_ALLOC | Could not allocate GPU device memory
MAGMA_ERR_HOST_ALLOC | Could not allocate CPU host memory
MAGMA_ERR_ILLEGAL_VALUE | An argument had an illegal value (deprecated; instead it should return -i to say the i-th argument was bad)
MAGMA_ERR_INVALID_PTR | Can't free pointer
MAGMA_ERR_NOT_IMPLEMENTED | Function or option not implemented
MAGMA_ERR_NOT_SUPPORTED | Function or option not supported on the current architecture

magma_xerbla() is called to report errors (mostly bad arguments) to the user.

magma_strerror() returns string description of an error code.

# Methodology

## One-sided matrix factorizations

The one-sided LU, Cholesky, and QR factorizations form a basis for solving linear systems. A general recommendation is to use LU for general n-by-n matrices, Cholesky for symmetric/Hermitian positive definite (SPD) matrices, and QR for solving least squares problems,

    min || A x - b ||

for general m-by-n, m > n matrices.

We use hybrid algorithms where the computation is split between the GPU and and the CPU. In general for the one-sided factorizations, the panels are factored on the CPU and the trailing sub-matrix updates on the GPU. Look-ahead techniques are used to overlap the CPU and GPU work and some communications.

In both the CPU and GPU interfaces the matrix to be factored resides in the GPU memory, and CPU-GPU transfers are associated only with the panels. The resulting matrix is accumulated (on the CPU or GPU according to the interface) along the computation, as a byproduct of the algorithm, vs. sending the the entire matrix when needed. In the CPU interface, the original transfer of the matrix to the GPU is overlapped with the factorization of the first panel. In this sense the CPU and GPU interfaces, although similar, are not derivatives of each other as they have different communication patterns.

Although the solution step has O(n) times less floating point operations than the factorization, it is still very important to optimize it. Solving a triangular system of equations can be very slow because the computation is bandwidth limited and naturally not parallel. Various approaches have been proposed in the past. We use an approach where diagonal blocks of A are explicitly inverted and used in a block algorithm. This results in a high performance, numerically stable algorithm, especially when used with triangular matrices coming from numerically stable factorization algorithms (e.g., as in LAPACK and MAGMA).

For instances when the GPU's single precision performance is much higher than its double precision performance, MAGMA provides a second set of solvers, based on the mixed precision iterative refinement technique. The solvers are based again on correspondingly the LU, QR, and Cholesky factorizations, and are designed to solve linear problems in double precision accuracy but at a speed that is characteristic for the much faster single precision computations. The idea is to use single precision for the bulk of the computation, namely the factorization step, and than use that factorization as a preconditioner in a simple iterative refinement process in double precision arithmetic. This often results in the desired high performance and high accuracy solvers.

## Two-sided matrix factorizations

As the one-sided matrix factorizations are the basis for various linear solvers, the two-sided matrix factorizations are the basis for eigen-solvers, and therefore form an important class of dense linear algebra routines. The two-sided factorizations have been traditionally more difficult to achieve high performance. The reason is that the two-sided factorizations involve large matrix-vector products which are memory bound, and as the gap between compute and communication power increases exponentially, these memory bound operations become an increasingly more difficult to handle bottleneck. GPUs though offer an attractive possibility to accelerate them. Indeed, having a high bandwidth (e.g. 10 times larger than current CPU bus bandwidths), GPUs can accelerate matrix-vector products significantly (10 to 30 times). Here, the panel factorization itself is hybrid, while the trailing matrix update is performed on the GPU.