Repository: pytorch
Issue: torch.linalg.qr is significantly slower on GPU compared to CPU and SVD for batched small matrices
Labels: module: cuda,triaged,module: linear algebra

### 🐛 Describe the bug

When performing QR decomposition on batched small matrices, torch.linalg.qr is significantly slower on the GPU compared to the CPU and even slower than torch.linalg.svd on the GPU. This behavior seems unexpected since QR decomposition is typically faster than SVD. Tested with pytorch 2.3 and pytorch 2.5 .

Steps to Reproduce:

```
import torch
import time

matrix = torch.randn(10, 30, 30, 30, 3, 3)

for device in ["cpu", "cuda"]:
    start = time.time()
    U, S, V = torch.linalg.svd(matrix.to(device))
    print("SVD Time:", time.time() - start, "Device:", device)
    
    start = time.time()
    Q, R = torch.linalg.qr(matrix.to(device))
    print("QR Time:", time.time() - start, "Device:", device)
```
Output on an Nvidia H100:
SVD Time: 0.5345005989074707   cpu
QR Time: 0.11194396018981934   cpu
SVD Time: 0.06591439247131348   cuda
QR Time: 8.115148305892944   cuda

Output on an Nvidia Rtx5000a:
SVD Time: 0.6414940357208252   cpu
QR Time: 0.11259078979492188   cpu
SVD Time: 0.8223276138305664   cuda
QR Time: 7.6162269115448   cuda



### Versions

PyTorch version: 2.5.1
Is debug build: False
CUDA used to build PyTorch: 12.4
ROCM used to build PyTorch: N/A

OS: Ubuntu 22.04.5 LTS (x86_64)
GCC version: (Ubuntu 11.4.0-1ubuntu1~22.04) 11.4.0
Clang version: Could not collect
CMake version: Could not collect
Libc version: glibc-2.35

Python version: 3.11.6 | packaged by conda-forge | (main, Oct  3 2023, 10:40:35) [GCC 12.3.0] (64-bit runtime)
Python platform: Linux-6.8.0-45-generic-x86_64-with-glibc2.35
Is CUDA available: True
CUDA runtime version: Could not collect
CUDA_MODULE_LOADING set to: LAZY
GPU models and configuration: 
GPU 0: NVIDIA H100 PCIe
GPU 1: NVIDIA H100 PCIe
GPU 2: NVIDIA H100 PCIe
GPU 3: NVIDIA H100 PCIe

Nvidia driver version: 550.90.07
cuDNN version: Could not collect
HIP runtime version: N/A
MIOpen runtime version: N/A
Is XNNPACK available: True

CPU:
Architecture:                         x86_64
CPU op-mode(s):                       32-bit, 64-bit
Address sizes:                        52 bits physical, 57 bits virtual
Byte Order:                           Little Endian
CPU(s):                               384
On-line CPU(s) list:                  0-383
Vendor ID:                            AuthenticAMD
Model name:                           AMD EPYC 9654 96-Core Processor
CPU family:                           25
Model:                                17
Thread(s) per core:                   2
Core(s) per socket:                   96
Socket(s):                            2
Stepping:                             1
Frequency boost:                      enabled
CPU max MHz:                          3707.8120
CPU min MHz:                          1500.0000
BogoMIPS:                             4799.78
Flags:                                fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt pdpe1gb rdtscp lm constant_tsc rep_good amd_lbr_v2 nopl nonstop_tsc cpuid extd_apicid aperfmperf rapl pni pclmulqdq monitor ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand lahf_lm cmp_legacy svm extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch osvw ibs skinit wdt tce topoext perfctr_core perfctr_nb bpext perfctr_llc mwaitx cpb cat_l3 cdp_l3 hw_pstate ssbd mba perfmon_v2 ibrs ibpb stibp ibrs_enhanced vmmcall fsgsbase bmi1 avx2 smep bmi2 erms invpcid cqm rdt_a avx512f avx512dq rdseed adx smap avx512ifma clflushopt clwb avx512cd sha_ni avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local user_shstk avx512_bf16 clzero irperf xsaveerptr rdpru wbnoinvd amd_ppin cppc arat npt lbrv svm_lock nrip_save tsc_scale vmcb_clean flushbyasid decodeassists pausefilter pfthreshold avic v_vmsave_vmload vgif x2avic v_spec_ctrl vnmi avx512vbmi umip pku ospke avx512_vbmi2 gfni vaes vpclmulqdq avx512_vnni avx512_bitalg avx512_vpopcntdq la57 rdpid overflow_recov succor smca fsrm flush_l1d debug_swap
Virtualization:                       AMD-V
L1d cache:                            6 MiB (192 instances)
L1i cache:                            6 MiB (192 instances)
L2 cache:                             192 MiB (192 instances)
L3 cache:                             768 MiB (24 instances)
NUMA node(s):                         2
NUMA node0 CPU(s):                    0-95,192-287
NUMA node1 CPU(s):                    96-191,288-383
Vulnerability Gather data sampling:   Not affected
Vulnerability Itlb multihit:          Not affected
Vulnerability L1tf:                   Not affected
Vulnerability Mds:                    Not affected
Vulnerability Meltdown:               Not affected
Vulnerability Mmio stale data:        Not affected
Vulnerability Reg file data sampling: Not affected
Vulnerability Retbleed:               Not affected
Vulnerability Spec rstack overflow:   Mitigation; Safe RET
Vulnerability Spec store bypass:      Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1:             Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2:             Mitigation; Enhanced / Automatic IBRS; IBPB conditional; STIBP always-on; RSB filling; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds:                  Not affected
Vulnerability Tsx async abort:        Not affected

Versions of relevant libraries:
[pip3] flake8==7.1.1
[pip3] mypy-extensions==1.0.0
[pip3] numpy==1.26.4
[pip3] numpydoc==1.8.0
[pip3] torch==2.5.1
[pip3] torchaudio==2.5.1
[pip3] torchvision==0.20.1
[pip3] triton==3.1.0
[conda] blas                      2.116                       mkl    conda-forge
[conda] blas-devel                3.9.0            16_linux64_mkl    conda-forge
[conda] ffmpeg                    4.3                  hf484d3e_0    pytorch
[conda] libblas                   3.9.0            16_linux64_mkl    conda-forge
[conda] libcblas                  3.9.0            16_linux64_mkl    conda-forge
[conda] libjpeg-turbo             2.0.0                h9bf148f_0    pytorch
[conda] liblapack                 3.9.0            16_linux64_mkl    conda-forge
[conda] liblapacke                3.9.0            16_linux64_mkl    conda-forge
[conda] mkl                       2022.1.0           h84fe81f_915    conda-forge
[conda] mkl-devel                 2022.1.0           ha770c72_916    conda-forge
[conda] mkl-include               2022.1.0           h84fe81f_915    conda-forge
[conda] numpy                     1.26.4          py311h64a7726_0    conda-forge
[conda] numpydoc                  1.8.0              pyhd8ed1ab_0    conda-forge
[conda] pytorch                   2.5.1           py3.11_cuda12.4_cudnn9.1.0_0    pytorch
[conda] pytorch-cuda              12.4                 hc786d27_7    pytorch
[conda] pytorch-mutex             1.0                        cuda    pytorch
[conda] torchaudio                2.5.1               py311_cu124    pytorch
[conda] torchtriton               3.1.0                     py311    pytorch
[conda] torchvision               0.20.1              py311_cu124    pytorch


cc @ptrblck @msaroufim @eqy @jianyuh @nikitaved @pearu @mruberry @walterddr @xwang233 @Lezcano