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The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence. The type of convergence is also discussed, which is global under an appropriate choice of the initial matrix. Numerical experiments are reported based on input matrices of different sizes as well as various stopping termination levels with comparisons to methods of the same nature and same number of matrix–matrix multiplications. The simulation results confirm the efficiency of the proposed scheme in contrast to its competitors of the same nature.

Key Clinical Message: Benign ancient retroperitoneal schwannomas (BARS) exhibit abdominal masses and flank pain to incidental findings at more advanced stages. Histopathological and immunohistochemical analysis is essential for confirmation of benign nature. Our patient was misdiagnosed as ureteric colic, highlighting the need to consider BARS in differential diagnosis to prevent complications like hydronephrosis. Ancient schwannomas are usually benign neoplasms that originate from Schwann cells of peripheral nerves. We present a novel case of a 24‐year‐old young male with left flank pain and nausea which was initially thought to be left ureteric colic. However, in‐depth imaging and biopsy revealed a retroperitoneal mass. The definitive diagnosis was narrowed down to Benign Retroperitoneal Ancient Schwannoma (BARS) via immunohistochemistry and histopathological analysis. This often marble‐shaped S100 protein‐positive tumor is an under‐recognized and potential cause of hydronephrosis if localized near the renal structures. In addition, the retroperitoneal location with infrarenal abdominal aortic adherence is another rare peculiarity in the present case that demands prompt diagnosis and surgical excision to avoid any cardiovascular sequelae such as hypotension and abdominal pain, as indicated by the natural history of growth of this benign tumor. Therefore, timely excision of this benign tumor prior to its further proliferation is paramount. We initially planned laparoscopic removal but adopted excision via laparotomy because of the proximity of the vital structures. The postoperative course of the patient was uneventful and subsequently the patient's presenting complaint of left abdominal flank pain greatly improved. The patient was advised to undergo follow‐up computed tomography scan of kidney ureter bladder and RFT evaluation 6 months postsurgery which indicated no evidence of recurrence or iatrogenic complications. The diagnosis and management of the present case share valuable experiences for similar future cases worldwide. [ABSTRACT FROM AUTHOR]