HLIBpro  2.5.1

The following parameters of 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 are defined in the HLIB::CFG namespace with corresponding sub namespaces.

You may either read/write the parameters within your source code, e.g.,

HLIB::CFG::BLAS::use_rrqr_trunc = true;

or via environment variables. In the latter case, "::" is replaced by "_" and "CFG" is omitted, e.g., instead of HLIB::CFG::BLAS::use_rrqr_trunc the environment variable HLIB_BLAS_use_rrqr_trunc is used with boolean values defined by 0 or 1.

Furthermore, a list of all parameters with values may be printed via


In the C bindings the corresponding functions for setting, reading and printing the parameters are

void hlib_set_config ( const char * option,
const char * value,
int * info );
void hlib_get_config ( const char * option,
char * value,
const size_t len,
int * info );

Please note, that the names and values correspond to the names and values of the environment variables and not of the C++ parameters.


Name Type Description Default Value
use_rrqr_trunc booluse RRQR based truncate instead of SVD based false
use_double_prec booluse double precision SVD for single precision types false
use_gesvj booluse gesvj instead of gesvd/gesdd false
gesvd_limit idx_tupper matrix size limit for using *gesvd (*gesdd for larger matrices) 1000
check_zeroes boolcheck for and remove zero rows/columns in input matrices of SVDfalse
check_inf_nan boolcheck for INF/NAN values in input data false


Rank-revealing QR (RRQR) uses column pivoted QR to determine the numerical rank of a given matrix. This is used in 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 to truncated low-rank matrices with respect to a given precision or rank. In contrast to the SVD based algorithm, RRQR does usually not compute a best approximate, which normally results in a larger rank of the low-rank matrices or reduced accuracy in case of a fixed rank. However, the RRQR algorithm is faster than SVD, which results in a better runtime of the ℋ-algorithm.

Please note, that this behaviour is strongly depended on the given ℋ-matrix. In extreme cases, the ℋ-arithmetic may be slower since the rank of low-rank matrices may be much higher compared to SVD.


If 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 is compiled with single precision value types, setting use_double_prec to true, results in using double precision for computing the SVD during truncation. This increases accuracy and may reduce memory due to more accurate rank estimates.


LAPACK provides an additional SVD algorithm (gesvj), which is often more accurate compared to the standard algorithms (gesvd, gesdd). Especially for extreme cases, e.g., very small singular values, this may be a better choice. However, gesvj is slower compared to the other algorithms (usually a factor of two).


The SVD based algorithm may have problems of the input matrix has zero rows/columns (more precise the LAPACK algorithms used by the truncation). To fix this, such rows/columns can be eliminated from the input matrices.


Name Type Description Default Value
nmin uint upper limit for minimal block size 60
split_mode default split mode during geometrical clusterung adaptive_split_axis
adjust_bbox bool adjust bounding box during clustering to local indices and not as defined by parent partitioning true
cluster_level_mode permit block clusters with clusters from different levels or not cluster_level_equal
sync_interface_depth bool synchronise depth of interface clusters with domain clusters in ND case true
sort_wrt_size bool sort sub clusters according to size, e.g. larger clusters first false


The optimal value for nmin depends on the given problem and what is to be optimised: memory or runtime. As a rule of thumb, the smaller nmin the less memory is used and the larger nmin the better the runtime.

The default value is chosen based on benchmarks for dense matrices. For sparse matrices a higher value is often better (nmin = 100 ... 300).


When partitioning a given geometry using binary space partitioning (BSP), the split axis may be chosen regularly (regular_split_axis), i.e., cycling through x, y and z axis or adpative (adaptive_split_axis), i.e., choose the longest axis. Theory usually assumes regular splitting while adaptive splitting often results in a better (less memory, better runtime) partitioning.


During BSP the sub domains may be assigned a bounding box by splitting the bounding box of the parent domain along the split axis or by recomputing the bounding box based on the coordinate information of the indices in the corresponding sub domains. The latter method usually reduces bounding box sizes and leads to a coarser partitioning of the ℋ-matrix, which in practise is often more optimal (less memory, better runtime). However, theory usually assumes a non-adaptive bounding box.


When building block clusters, i.e., a product of two clusters, normally only clusters of the same level are permitted to form block clusters. For some applications it may result in a better partitioning when also allowing clusters from different levels during block cluster tree construction (see also sync_interface_depth).


If nested dissection is used, the depth of the interface tree is normally adjusted to the depth of the domain trees. For some applications, switching this off may result in a better partitioning (see also cluster_level_mode).


Name Type Description Default Value
use_dag bool use DAG based arithmetic true
abs_eps real default absolute error bound for low-rank SVD 0
recompress bool recompress matrix after low-rank approximation true
use_zero_matrix bool use special zero matrix type for domain-domain blocks true
build_ghost_matrix bool build ghost matrices for remote blocks false
build_sparse_matrix bool use sparse matrices during construction false
eval_type point-wise or block-wise evaluation of blocks block_wise
storage_type storage type of diagonal blocks in factorisation store_inverse
max_seq_size size_t switching point from parallel to sequential mode 100
max_seq_size_vec size_t switching point for vector based operations 250
coarsen_build bool coarsen matrix during construction false
coarsen_arith bool coarsen matrix during arithmetic false
check_cb_ret bool check data returned by callback functions during construction true
arith_max_check uint upper matrix size limit for addition tests during arithmetic 0


For multi and many core chips, the DAG based ℋ-arithmetic is able to utilise the CPU resources much better than the classical recursive algorithms. Only for sequential computations, a slightly better runtime may be possible with the recursive approach due to overhead for DAG based computations.


Singular values smaller than abs_eps are removed from the resulting matrix during truncation independent on the given accuracy or rank.


The low-rank approximations computed during ℋ-matrix construction may not be optimal with respect to the rank. A recompression using SVD (or RRQR) may reduce the rank and hence memorr without affecting accuracy.


When using nested dissection, off-diagonal domain-domain blocks remain zero during ℋ-LU factorisation. A special matrix type can therefore be used to signal this property to the ℋ-arithmetic and to improve performance of the algorithms.

However, make sure that only such ℋ-algorithms are used, which preserve the zero property. Otherwise, an error may occur.


The evaluation type affects the handling of diagonal leaf blocks. Either factorisation is also performed for such blocks (point_wise) or the blocks are handled as a full block using inversion (block_wise). The latter has several advantages, e.g., permits limited pivoting during inversion and increases performance due to better cache utilisation (BLAS level 3 functions).


When computing matrix factorisations, the diagonal leaf blocks may store either the actual result of the factorisation (store_normal) or the inverse of such blocks (store_inverse). Since for evaluation of the inverse operators, either during factorisation or during matrix-vector multiplication, the inverse if the matrix blocks is needed, storing the inverse results in better performance.

However, if also the original matrix should be evaluated using the factorised form, normal storage may be more optimal in terms of runtime.

max_seq_size (_vec)

Matrices smaller than max_seq_size are handled by ℋ-arithmetic sequentially since this is more efficient compared to parallel handling. Same holds for max_seq_size_vec in case of matrix-vector operations.


Name Type Description Default Value
max_iter uint maximal number of iterations 100
rel_res_red real relative residual reduction, e.g., stop if $ |r_n| / |r_0| < \varepsilon$ 1e-8
abs_res_red real absolute residual reduction, e.g., stop if $|r_n| < \varepsilon$ 1e-14
rel_res_growth real relative residual growth (divergence), e.g., stop if $|r_n| / |r_0| > \varepsilon$ 1e6
gmres_restart uint restart for GMRES iteration 20
initialise_start_value bool initialise start value before iteration true
use_exact_residual bool compute exact residual during iteration false


This parameter stops the iteration in case of divergence.


Normally, the start value of the iteration is initialised as $x_0 := W b$ with $W$ being the preconditioner. In case of a good (or very good) preconditioner, e.g., $W \sim A^{-1}$, this may eliminate iteration at all.

However, if the user has a good guess for the start value this behaviour may be switched off.


Most iterations approximate the residual norm during computations. Especially in the preconditioned case, this may result in a large deviation from the real residual norm. At the expense of one (unpreconditioned case) or two (preconditioned case) additional matrix-vector multiplications, the exact norm is computed instead.


Name Type Description Default Value
quad_order uint default quadrature order 4
adaptive_quad_order bool use distance adaptive quadrature order true
use_simd bool use special vector functions if available true
use_simd_sse3 bool use special SSE3 vector functions if available true
use_simd_avx bool use special AVX vector functions if available true
use_simd_avx2 bool use special AVX2 vector functions if available true
use_simd_mic bool use special MIC vector functions if available true


For the evaluation of the bilinear forms a quadrature rule is used for which the default order is defined with this parameter. For high accuracy ℋ-arithmetic, a higher order may be used, e.g., 5 or 6.


For far-field evaluation, the quadrature order may be reduced for normal bilinear forms, resulting in much reduced runtime.


All implemented bilinear forms have special implementations using SIMD instructions of the CPU. Normally, this results in a faster computation compared to compiler generated code. So, in most cases this flag is only needed for comparison or for debugging.


Name Type Description Default Value
use_matlab_syntax bool use Matlab syntax for printing complex numbers, vectors and stdio off
color_mode use color in terminal output if supported true
charset_mode use ascii or unicode in normal terminal output depends on terminal