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cupolar_halley.cu
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cupolar_halley.cu
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/* MIT License
*
* Copyright (c) 2024 Maximilian Behr
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
* SOFTWARE.
*/
#include <cuComplex.h>
#include <cuda_runtime.h>
#include <math.h>
#include <stdint.h>
#include <thrust/device_ptr.h>
#include <thrust/inner_product.h>
#include "checkcuda.h"
#include "cupolar.h"
#include "cupolar_frobenius.h"
#include "cupolar_traits.h"
const static cusolverAlgMode_t CUSOLVER_ALG = CUSOLVER_ALG_0;
template <typename T>
static int cupolar_HalleyBufferSize(const int n, size_t *d_bufferSize, size_t *h_bufferSize) {
/*-----------------------------------------------------------------------------
* initialize with zero
*-----------------------------------------------------------------------------*/
*d_bufferSize = 0;
*h_bufferSize = 0;
/*-----------------------------------------------------------------------------
* get device and host workspace size for LU factorization
*-----------------------------------------------------------------------------*/
// create cusolver handle
cusolverDnHandle_t cusolverH;
CHECK_CUSOLVER(cusolverDnCreate(&cusolverH));
// create cusolver params
cusolverDnParams_t params;
CHECK_CUSOLVER(cusolverDnCreateParams(¶ms));
CHECK_CUSOLVER(cusolverDnSetAdvOptions(params, CUSOLVERDN_GETRF, CUSOLVER_ALG));
// compute workspace size
CHECK_CUSOLVER(cusolverDnXgetrf_bufferSize(cusolverH, params, n, n, cupolar_traits<T>::dataType, nullptr, n, cupolar_traits<T>::computeType, d_bufferSize, h_bufferSize));
// free workspace
CHECK_CUSOLVER(cusolverDnDestroy(cusolverH));
CHECK_CUSOLVER(cusolverDnDestroyParams(params));
/*-----------------------------------------------------------------------------
* compute final workspace size
*-----------------------------------------------------------------------------*/
*d_bufferSize += sizeof(T) * n * n * 3 + sizeof(int64_t) * n + sizeof(int);
return 0;
}
int cupolar_sHalleyBufferSize(const int n, size_t *d_bufferSize, size_t *h_bufferSize) {
return cupolar_HalleyBufferSize<float>(n, d_bufferSize, h_bufferSize);
}
int cupolar_dHalleyBufferSize(const int n, size_t *d_bufferSize, size_t *h_bufferSize) {
return cupolar_HalleyBufferSize<double>(n, d_bufferSize, h_bufferSize);
}
int cupolar_cHalleyBufferSize(const int n, size_t *d_bufferSize, size_t *h_bufferSize) {
return cupolar_HalleyBufferSize<cuComplex>(n, d_bufferSize, h_bufferSize);
}
int cupolar_zHalleyBufferSize(const int n, size_t *d_bufferSize, size_t *h_bufferSize) {
return cupolar_HalleyBufferSize<cuDoubleComplex>(n, d_bufferSize, h_bufferSize);
}
__device__ inline static cuComplex operator*(float a, const cuComplex &b) {
return make_cuComplex(a * b.x, a * b.y);
}
__device__ inline static cuComplex operator+(float a, const cuComplex &b) {
return make_cuComplex(a + b.x, b.y);
}
__device__ inline static cuDoubleComplex operator*(double a, const cuDoubleComplex &b) {
return make_cuDoubleComplex(a * b.x, a * b.y);
}
__device__ inline static cuDoubleComplex operator+(double a, const cuDoubleComplex &b) {
return make_cuDoubleComplex(a + b.x, b.y);
}
template <typename S, typename T>
__global__ void prepare_Halley(const int n, const S a, const S b, const S c, T *QHQ, T *aIbQHQ) {
int i0 = blockIdx.x * blockDim.x + threadIdx.x;
int j0 = blockIdx.y * blockDim.y + threadIdx.y;
for (int j = j0; j < n; j += gridDim.y * blockDim.y) {
for (int i = i0; i < n; i += gridDim.x * blockDim.x) {
T tmp = QHQ[i + j * n];
if (i == j) {
QHQ[i + j * n] = S{1.0} + c * tmp;
aIbQHQ[i + j * n] = a + b * tmp;
} else {
QHQ[i + j * n] = c * tmp;
aIbQHQ[i + j * n] = b * tmp;
}
}
}
}
template <typename T>
__global__ void identity(const int n, T *A, const int lda) {
int i0 = blockIdx.x * blockDim.x + threadIdx.x;
int j0 = blockIdx.y * blockDim.y + threadIdx.y;
for (int j = j0; j < n; j += gridDim.y * blockDim.y) {
for (int i = i0; i < n; i += gridDim.x * blockDim.x) {
A[i + j * lda] = (i == j) ? cupolar_traits<T>::one : cupolar_traits<T>::zero;
}
}
}
template <typename T>
static int cupolar_Halley(const int n, const T *A, void *d_buffer, void *h_buffer, T *Q, T *H) {
/*-----------------------------------------------------------------------------
* derived types
*-----------------------------------------------------------------------------*/
using S = typename cupolar_traits<T>::S; // real type: double for cuDoubleComplex, float for cuComplex
/*-----------------------------------------------------------------------------
* constants and variables
*-----------------------------------------------------------------------------*/
int ret = 0, iter = 1;
constexpr int maxiter = 100;
const S tol = std::cbrt(std::numeric_limits<S>::epsilon()); // square root of machine epsilon - newton iteration converges quadratically
S alpha, beta, l, a, b, c;
size_t lworkdevice = 0, lworkhost = 0;
/*-----------------------------------------------------------------------------
* create cuBlas handle
*-----------------------------------------------------------------------------*/
cublasHandle_t cublasH;
CHECK_CUBLAS(cublasCreate(&cublasH));
/*-----------------------------------------------------------------------------
* create cusolver handle and params structure
*-----------------------------------------------------------------------------*/
cusolverDnHandle_t cusolverH;
CHECK_CUSOLVER(cusolverDnCreate(&cusolverH));
cusolverDnParams_t params;
CHECK_CUSOLVER(cusolverDnCreateParams(¶ms));
CHECK_CUSOLVER(cusolverDnSetAdvOptions(params, CUSOLVERDN_GETRF, CUSOLVER_ALG));
/*-----------------------------------------------------------------------------
* split memory buffer
* memory layout: |Qold, Qtmp, Qtmp2, ipiv, info, d_work|
*-----------------------------------------------------------------------------*/
T *Qold = reinterpret_cast<T *>(d_buffer);
T *Qtmp = reinterpret_cast<T *>(Qold + n * n); // put Qtmp after Qold
T *Qtmp2 = reinterpret_cast<T *>(Qtmp + n * n); // put Qtmp2 after Qold
int64_t *d_ipiv = reinterpret_cast<int64_t *>(Qtmp2 + n * n); // put d_ipiv after Qtmp2
int *d_info = reinterpret_cast<int *>(d_ipiv + n); // put d_info after d_ipiv
void *d_work = reinterpret_cast<int *>(d_info + 1); // put d_work after d_info
void *h_work = reinterpret_cast<void *>(h_buffer);
std::swap(Q, Qold);
/*-----------------------------------------------------------------------------
* compute alpha = || A ||_F
*-----------------------------------------------------------------------------*/
CHECK_CUPOLAR(cupolar_normFro(n, n, A, &alpha));
/*-----------------------------------------------------------------------------
* compute A^-1
*-----------------------------------------------------------------------------*/
// copy A to Qtmp
CHECK_CUDA(cudaMemcpy(Qtmp, A, sizeof(T) * n * n, cudaMemcpyDeviceToDevice));
// workspace query for LU factorization
CHECK_CUSOLVER(cusolverDnXgetrf_bufferSize(cusolverH, params, n, n, cupolar_traits<T>::dataType, Qtmp, n, cupolar_traits<T>::computeType, &lworkdevice, &lworkhost));
// compute LU factorization and set right side to identity on different streams
cudaStream_t streamLU, streamIdentity;
CHECK_CUDA(cudaStreamCreate(&streamLU));
CHECK_CUDA(cudaStreamCreate(&streamIdentity));
CHECK_CUSOLVER(cusolverDnSetStream(cusolverH, streamLU));
CHECK_CUSOLVER(cusolverDnXgetrf(cusolverH, params, n, n, cupolar_traits<T>::dataType, Qtmp, n, d_ipiv, cupolar_traits<T>::computeType, d_work, lworkdevice, h_work, lworkhost, d_info));
// set right-hand side to identity
{
dim3 grid((n + 15) / 16, (n + 15) / 16);
dim3 block(16, 16);
identity<<<grid, block>>>(n, Qtmp2, n);
CHECK_CUDA(cudaPeekAtLastError());
}
// synchronize and destroy streams
CHECK_CUDA(cudaStreamSynchronize(streamLU));
CHECK_CUDA(cudaStreamSynchronize(streamIdentity));
CHECK_CUDA(cudaStreamDestroy(streamLU));
CHECK_CUDA(cudaStreamDestroy(streamIdentity));
CHECK_CUSOLVER(cusolverDnSetStream(cusolverH, 0));
// solve the linear system to compute the hermitian/transposed inverse of Qold
CHECK_CUSOLVER(cusolverDnXgetrs(cusolverH, params, CUBLAS_OP_N, n, n, cupolar_traits<T>::dataType, Qtmp, n, d_ipiv, cupolar_traits<T>::computeType, Qtmp2, n, d_info));
/*-----------------------------------------------------------------------------
* compute beta = 1 / || A^-1 ||_F
*-----------------------------------------------------------------------------*/
CHECK_CUPOLAR(cupolar_normFro(n, n, Qtmp2, &beta));
beta = S{1.0} / beta;
/*-----------------------------------------------------------------------------
* compute l for the first iteration
*-----------------------------------------------------------------------------*/
l = beta / alpha;
/*-----------------------------------------------------------------------------
* copy A to Qold and scale A/alpha = Qold / alpha
*-----------------------------------------------------------------------------*/
CHECK_CUDA(cudaMemcpy(Qold, A, sizeof(T) * n * n, cudaMemcpyDeviceToDevice));
{
S alphainv = S{1.0} / alpha;
CHECK_CUBLAS(cupolar_traits<T>::cublasXdscal(cublasH, n * n, &alphainv, Qold, 1));
}
/*-----------------------------------------------------------------------------
* Halley iteration
*-----------------------------------------------------------------------------*/
iter = 1;
static_assert(maxiter >= 1, "maxiter >= 1");
while (true) {
/*-----------------------------------------------------------------------------
* update scaling parameters
*-----------------------------------------------------------------------------*/
{
S d = std::pow(S{4.0} * (S{1.0} - l * l) / (l * l * l * l), S{1.0} / S{3.0});
a = std::sqrt(S{1.0} + d) + S{0.5} * std::sqrt(S{8.0} - S{4.0} * d + S{8.0} * (S{2.0} - l * l) / (l * l * std::sqrt(S{1.0} + d)));
b = (a - S{1.0}) * (a - S{1.0}) / S{4.0};
c = a + b - S{1.0};
l = l * (a + b * l * l) / (S{1.0} + c * l * l);
}
/*-----------------------------------------------------------------------------
* compute Qold^H Qold -> Qtmp
*-----------------------------------------------------------------------------*/
CHECK_CUBLAS(cupolar_traits<T>::cublasXgemm(cublasH, CUBLAS_OP_C, CUBLAS_OP_N, n, n, n, &cupolar_traits<T>::one, Qold, n, Qold, n, &cupolar_traits<T>::zero, Qtmp, n));
/*-----------------------------------------------------------------------------
* compute I + c * Qtmp -> Qtmp and a * I + b * Qtmp -> Qtmp2
*-----------------------------------------------------------------------------*/
{
dim3 grid((n + 15) / 16, (n + 15) / 16);
dim3 block(16, 16);
prepare_Halley<<<grid, block>>>(n, a, b, c, Qtmp, Qtmp2);
CHECK_CUDA(cudaPeekAtLastError());
}
/*-----------------------------------------------------------------------------
* solve the system (I + c * Qold^H*Qold)\(a * I + b * Qold^H*Qold)
*-----------------------------------------------------------------------------*/
// workspace query for LU factorization
CHECK_CUSOLVER(cusolverDnXgetrf_bufferSize(cusolverH, params, n, n, cupolar_traits<T>::dataType, Qtmp, n, cupolar_traits<T>::computeType, &lworkdevice, &lworkhost));
// compute LU factorization
CHECK_CUSOLVER(cusolverDnXgetrf(cusolverH, params, n, n, cupolar_traits<T>::dataType, Qtmp, n, d_ipiv, cupolar_traits<T>::computeType, d_work, lworkdevice, h_work, lworkhost, d_info));
// solve the linear system
CHECK_CUSOLVER(cusolverDnXgetrs(cusolverH, params, CUBLAS_OP_N, n, n, cupolar_traits<T>::dataType, Qtmp, n, d_ipiv, cupolar_traits<T>::computeType, Qtmp2, n, d_info));
/*-----------------------------------------------------------------------------
* update Q as Q <- Qold*(I + c * Qold^H*Qold)\(a * I + b * Qold^H*Qold)
*-----------------------------------------------------------------------------*/
CHECK_CUBLAS(cupolar_traits<T>::cublasXgemm(cublasH, CUBLAS_OP_N, CUBLAS_OP_N, n, n, n, &cupolar_traits<T>::one, Qold, n, Qtmp2, n, &cupolar_traits<T>::zero, Q, n));
/*-----------------------------------------------------------------------------
* compute relative change of Q and Qold
*-----------------------------------------------------------------------------*/
S diffQQold, nrmQ;
CHECK_CUPOLAR(cupolar_diffnormFro(n, n, Q, Qold, &diffQQold));
CHECK_CUPOLAR(cupolar_normFro(n, n, Q, &nrmQ));
// printf("iter=%d, diffQQold=%e, nrmQ=%e, rel. change=%e\n", iter, diffQQold, nrmQ, diffQQold / nrmQ);
/*-----------------------------------------------------------------------------
* stopping criteria
*-----------------------------------------------------------------------------*/
// relative change of Q and Qold is smaller than tolerance
if (diffQQold < nrmQ * tol) {
break;
}
// maximum number of iterations reached
if (iter == maxiter) {
fprintf(stderr, "%s-%s:%d no convergence - maximum number of iterations reached\n", __func__, __FILE__, __LINE__);
fflush(stderr);
ret = -1;
break;
}
/*-----------------------------------------------------------------------------
* swap Q and Qold for the next iteration
*-----------------------------------------------------------------------------*/
std::swap(Q, Qold);
iter++;
}
/*-----------------------------------------------------------------------------
* copy Q and Qold if necessary
*-----------------------------------------------------------------------------*/
if (iter % 2 == 1) {
CHECK_CUDA(cudaMemcpy(Qold, Q, sizeof(T) * n * n, cudaMemcpyDeviceToDevice));
}
/*-----------------------------------------------------------------------------
* compute H if wanted
*-----------------------------------------------------------------------------*/
if (H) {
// compute H = Qold^H * A
CHECK_CUBLAS(cupolar_traits<T>::cublasXgemm(cublasH, CUBLAS_OP_C, CUBLAS_OP_N, n, n, n, &cupolar_traits<T>::one, Q, n, A, n, &cupolar_traits<T>::zero, Qtmp, n));
// correct symmetry/hermitianess of H
CHECK_CUBLAS(cupolar_traits<T>::cublasXgeam(cublasH, CUBLAS_OP_N, CUBLAS_OP_C, n, n, &cupolar_traits<T>::half, Qtmp, n, &cupolar_traits<T>::half, Qtmp, n, H, n));
}
/*-----------------------------------------------------------------------------
* destroy cuBlas and cuSolver handle and params structure
*-----------------------------------------------------------------------------*/
CHECK_CUSOLVER(cusolverDnDestroyParams(params));
CHECK_CUSOLVER(cusolverDnDestroy(cusolverH));
CHECK_CUBLAS(cublasDestroy(cublasH));
/*-----------------------------------------------------------------------------
* return
*-----------------------------------------------------------------------------*/
return ret;
}
int cupolar_sHalley(const int n, const float *A, void *d_buffer, void *h_buffer, float *Q, float *H) {
return cupolar_Halley(n, A, d_buffer, h_buffer, Q, H);
}
int cupolar_dHalley(const int n, const double *A, void *d_buffer, void *h_buffer, double *Q, double *H) {
return cupolar_Halley(n, A, d_buffer, h_buffer, Q, H);
}
int cupolar_cHalley(const int n, const cuComplex *A, void *d_buffer, void *h_buffer, cuComplex *Q, cuComplex *H) {
return cupolar_Halley(n, A, d_buffer, h_buffer, Q, H);
}
int cupolar_zHalley(const int n, const cuDoubleComplex *A, void *d_buffer, void *h_buffer, cuDoubleComplex *Q, cuDoubleComplex *H) {
return cupolar_Halley(n, A, d_buffer, h_buffer, Q, H);
}