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In order to overcome the computational challenges associated with block preconditioners for Krylov subspace methods, particularly those arising from Schur complement systems, this paper proposes an improved new block (INB) preconditioner for solving 3 × 3 block saddle point problems. A detailed semi-convergence analysis of the iterative scheme induced by the INB preconditioner is provided. Moreover, the spectral properties of the preconditioned matrix are analyzed, revealing strong eigenvalue clustering around one. Efficient formulas for selecting quasi-optimal parameters are derived based on Frobenius-norm minimization. Extensive numerical experiments demonstrate that the proposed INB preconditioner significantly reduces iteration counts and CPU time compared with several existing block preconditioners.

In this paper, inspired by the simplified relaxed alternating positive semi-definite splitting preconditioner (Xiong and Li in J Appl Math Comput, 2023), a modified preconditioner is established to solve the three-by-three block saddle point problems. The advantage of the modified preconditioner is that its parameters can be easy to select, which can avoid the work of the calculation of the parameters. Next, the spectral property of the preconditioned matrix with modified preconditioner is also discussed. In the end, three examples are provided to show the effectiveness of the modified preconditioner.

Objective: To compare the outcome of Perianal block versus Saddle block in the treatment of anal fissure using lateral sphincterotomy procedure. Study Design: Quasi-experimental study. Place and Duration of Study: Combined Military Hospital, Multan Pakistan, from Jun 2022 to Mar 2023. Methodology: The sample of 60 patients with anal fissures were randomly divided in two equal Groups (Group-A and B) by lottery method. Group-A patients underwent perianal block while Group-B patients received Saddle block. Both the Groups underwent lateral sphincterotomy procedure for the treatment of anal fissure. Pain assessment was done every hour for first 6 hours and then after 2 hours for the next 24 hours, and total pain-free period was defined as the mean time taken from administration of block till the time the patient required rescue analgesia (75 mg diclofenac sodium intramuscularly). Results: Males were (38)63.3%, whereas (22)36.7% were females. Mean age of these patients was 36.73±9.19 years (range; 22–58 years), and (43)71.7% were aged up to 40 years. Mean time taken for rescue analgesia in Group-A was 288.17±58.13 minutes versus 117.50±18.18 minutes in Group-B (p<0.001). Conclusion:  Mean time taken for rescue analgesia was significantly higher in perianal block as compared with saddle block. Hence, the use of a perianal block reduces additional use of analgesic drugs.

For solving the saddle point linear systems with three-by-three block structure, in this paper, based on the alternating positive semi-definite splitting (APSS) preconditioner developed by Aslani et al. (Filomat, 15:5181–5194, 2021), we presented a new preconditioner called simplified relaxed APSS (SRAPSS) preconditioner. Spectral properties of the SRAPSS preconditioned matrix are analyzed. Numerical experiments verify that the proposed preconditioner is effective compared with some existing preconditioners.

In this paper, to solve the three-by-three block saddle-point problem, a new block triangular (NBT) preconditioner is established, which can effectively avoid the solving difficulty that the coefficient matrices of linear subsystems are Schur complement matrices when the block preconditioner is applied to the Krylov subspace method. Theoretical analysis shows that the iteration method produced by the NBT preconditioner is unconditionally convergent. Besides, some spectral properties are also discussed. Finally, numerical experiments are provided to show the effectiveness of the NBT preconditioner.

Background Despite the benefits attributed to the use of local anesthesia (LA) for open hemorrhoidectomy (OH) in developed countries, this technique is still not considered as the first line technique in low-income countries such as Uganda; therefore, we aimed at comparing the cost of OH under LA versus Saddle block among patients with 3rd or 4th degree hemorrhoids. Methods This trial was conducted from December 2021 to May 2022 among patients with primary uncomplicated 3rd or 4th degree hemorrhoids. The operating time, and direct costs in (US$) including medical and non-medical were recorded. We analysed the cost in the two groups (local anesthesia versus saddle block) using SPSS version 23.0. Results Findings of fifty-eight patients were analysed including 29 participants per group. There was a significant difference in operating time and cost among the two groups (p < 0.05). The mean operating time was 15.52 ± 5.34(SD) minutes versus 33.72 ± 11.54 min for OH under LA and SB respectively. The mean cost of OH under LA was 57.42 ± 8.90 US$ compared to 63.38 ± 12.77US$ in SB group. Conclusion The use of local anesthesia for OH was found to have less operating time with high-cost effectiveness. Being affordable, local anesthesia can help to increase the turnover of patients who would otherwise wait for the availability of anesthesia provider. Policy makers should emphasize its applicability in low-income settings to help in the achievement of 2030 global surgery goals. Trial registration Pan African Clinical Trials Registry, PACTR202110667430356. Registered on 08/10/2021.

Background: Preemptive analgesics are commonly used to increase analgesic efficacy and patient satisfaction. The aim of this study was to evaluate the preemptive analgesic effect of intramuscular diclofenac on minimal effective dose spinal anesthesia for perianal surgeries. Materials and Methods: Fifty patients ASA I&II were divided randomly into two groups, control group (GC N = 25) and Diclofenac group (GD N = 25), both groups received saddle block with 5% hyperbaric bupivacaine 0.5 mL (2.5 mg). Thirty minutes before the saddle block, patients in GD received 75 mg (3 mL) diclofenac intramuscularly, whereas patients in GC received 3 mL saline intramuscularly. The differences in the time for the first analgesic request, postoperative analgesic consumption as well as, visual analog scale, were our primary outcomes. Results: Fifty patients (25 in each group) undergoing perianal surgery completed the study successfully. The time to the first request of analgesia was significantly longer in GD 511.8 (108.07) min. compared to the GC 179.56 (49.24) min with P = 0.00001, as well as the total consumption of rescue analgesic (tramadol hydrochloride) was significantly less in GD 66 (23.8) mg compared to 104 (28.5) mg in the GC with P = 0.00001. Conclusion: Preemptive intramuscular diclofenac sodium with minimal dose bupivacaine saddle block significantly minimized the postoperative analgesic consumption and delayed the first analgesia request after perianal surgery.

Background While open hemorrhoidectomy under local anesthesia has been shown to be more cost-effective with shorter operation times and lower complication rates, local anesthesia is still not considered as a first-line technique in low-income countries like Uganda. The objective of this trial is to compare open hemorrhoidectomy using local anesthesia versus saddle block among patients with primary uncomplicated 3rd- or 4th-degree hemorrhoids in western Uganda. Methods The protocol for a prospective equivalence randomized, double-blind controlled trial was conducted among patients with primary uncomplicated 3rd- or 4th-degree hemorrhoids. Recruitment was started in December 2021 and is expected to end in May 2022. Consenting participants who require open hemorrhoidectomy indicated at Kampala International Teaching Hospital, Uganda, will be randomized into two groups of 29 patients per arm. Discussion The primary outcome of this study is to compare the occurrences of postoperative pain following open hemorrhoidectomy using the visual analog scale in an interval of 2, 4, and 6 h and 7 days postoperatively. Furthermore, the mean operative time from the induction of anesthesia to the end of the surgical procedure as well as the cost-effectiveness of the 2 techniques will be assessed in both groups. Open hemorrhoidectomy under local anesthesia has the potential to offer benefits to patients but most importantly expediting return to baseline and functional status, shorter hospital stay by meeting the faster discharge criteria, and reduction in costs associated with reduced length of stay and complications. Trial registration Pan African Clinical Trials Registry PACTR202110667430356. Registered on 8 October 2021

Background There is disparity in evidence on pain assessment post open hemorrhoidectomy (OH) using local anesthesia and its use in developing countries compared to developed countries. Therefore, we conducted this study to assess the occurrence of postoperative pain following open hemorrhoidectomy under local anesthesia versus saddle block for uncomplicated 3 rd or 4 th degree hemorrhoids. Methods This was a prospective equivalence randomized, double blind controlled trial conducted from December 2021 to May 2022 among patients with primary uncomplicated 3 rd or 4 th degree hemorrhoids. Pain severity was assessed at 2, 4 and 6 h post open hemorrhoidectomy using visual analogue scale (VAS). Data was analysed using SPSS version 26 at a p  < 0.05 as statically significant using visual analogue scale (VAS). Results We recruited 58 participants in this study who underwent open hemorrhoidectomy under local anesthesia or saddle block (29 participants per group). The sex ratio was of 1.15 of female to male and a mean age of 39 ± 13. VAS was found to be different at 2 h post OH compare to other time of pain assessment but not statically significant by area under the cover (AUC) (95% CI = 486–0.773: AUC = 0.63; p  = 0.09) with a none significance by Kruskal–Wallis’s test (p:0.925). Conclusion Local anesthesia was found to be having a similar pain severity occurrence in post operative period among patients undergoing open hemorrhoidectomy for primary uncomplicated 3 rd or 4 th degree hemorrhoids. Close monitoring of pain in postoperative period is mandatory especially at 2 h to assess need of analgesia. Trial registration Pan African Clinical Trials Registry, PACTR202110667430356. Registered on 8 th October, 2021.

Spinal anesthesia is the technique of choice in transurethral resection of prostate (TURP). The major complication of spinal technique is risk of hypotension. Saddle block paralyzed pelvic muscles and sacral nerve roots and hemodynamic derangement is less. To compare the hemodynamic changes and adequate surgical condition between saddle block and subarachnoid block for TURP. Ninety patients of aged between 50 to 70 years of ASA-PS I, II scheduled for TURP were randomly allocated into 2 groups of 45 in each group. Group A patients were received spinal (2 ml of hyperbaric bupivacaine) and Group B were received saddle block (2 ml of hyperbaric bupivacaine). Baseline systolic, diastolic and mean arterial pressure, heart rate, oxygen saturation were recorded and measured subsequently. The height of block was noted in both groups. Hypotension was corrected by administration of phenylephrine 50 mcg bolus and total requirement of vasopressor was noted. Complications (volume overload, TURP syndrome etc.) were noted. Incidence of hypotension and vasopressor requirement was less (P < 0.01) in Gr B patients. Adequate surgical condition was achieved in both groups. There was no incidence of volume overload, TURP syndrome, and bladder perforation. TURP can be safely performed under saddle block without hypotension and less vasopressor requirement.