SVD is an initialism that may refer to several topics in different fields, including:
[[Singular value decomposition]], a matrix factorization technique in linear algebra (see Mathematics section)
[[Spontaneous vaginal delivery]], a method of childbirth (see Medicine section)
[[SVD (rifle)]], a Soviet semi-automatic sniper rifle (see Weapons section)
For other uses, see the Other uses section.
Mathematics
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization technique that decomposes any real or complex matrix A \in \mathbb{C}^{m \times n} into the product of three matrices: A = U \Sigma V^H, where U \in \mathbb{C}^{m \times m} is a unitary matrix (satisfying U^H U = I), \Sigma \in \mathbb{R}^{m \times n} is a rectangular diagonal matrix with non-negative real entries on its main diagonal known as singular values, and V^H is the conjugate transpose (Hermitian adjoint) of a unitary matrix V \in \mathbb{C}^{n \times n}.[1] This decomposition exists and is unique up to signs in the columns of U and V for equal singular values.[2]The singular values \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_p \geq 0, where p = \min(m, n), are arranged in non-increasing order along the diagonal of \Sigma, with all off-diagonal elements zero.[3] For the real case, the formula simplifies to A = U \Sigma V^T with orthogonal matrices U and V.[4]Key properties of the SVD include the orthogonality of the columns of U and V, which form orthonormal bases for \mathbb{C}^m and \mathbb{C}^n, respectively; the non-negativity of the singular values; and the fact that the rank of A equals the number of positive singular values.[5] Additionally, the squared singular values are the eigenvalues of A^H A or A A^H, and the columns of V corresponding to non-zero singular values span the row space of A.[6]Geometrically, the SVD interprets the linear transformation represented by A as a composition of three operations: a rotation or reflection by V^H, followed by anisotropic scaling along the coordinate axes by \Sigma, and finally another rotation or reflection by U.[7] This reveals the intrinsic geometry of the transformation, such as stretching factors given by the singular values.[8]The SVD for square matrices was first derived by Eugenio Beltrami in 1873 and independently by Camille Jordan in 1874, initially for simplifying quadratic forms.[9] Its generalization to rectangular matrices was established in the 1930s by Carl Eckart and Gale Young, who connected it to best approximations in the least-squares sense.[10]Computationally, the standard algorithm for the full SVD is the Golub-Reinsch method, which first reduces A to upper bidiagonal form using Householder transformations and then applies the QR algorithm to compute the singular values and vectors, achieving a time complexity of O(\min(m n^2, m^2 n)).[11] This approach remains foundational in numerical linear algebra libraries.[12]
Generalized singular value decomposition
The generalized singular value decomposition (GSVD) provides an extension of the singular value decomposition to pairs of matrices, typically A \in \mathbb{R}^{m \times p} and B \in \mathbb{R}^{n \times p} sharing the same number of columns. In the standard formulation, it factorizes these matrices as A = U \Sigma X^{-1} and B = V M X^{-1}, where U \in \mathbb{R}^{m \times m} and V \in \mathbb{R}^{n \times n} are orthogonal matrices, X \in \mathbb{R}^{p \times p} is nonsingular, and \Sigma and M are diagonal (or block-diagonal) matrices whose diagonal elements \alpha_i and \beta_i define the generalized singular values k_i = \alpha_i / \beta_i \geq 0.[13] For indices i up to \min(m, n, p), these satisfy \alpha_i^2 + \beta_i^2 = 1 in the non-trivial part, with structures of \Sigma and M accommodating rectangular dimensions through zero-padded blocks. The ratios k_i are non-negative real numbers. A variant exists for matrices sharing rows, but the column-sharing form is conventional.Key properties of the GSVD include its ability to handle cases where B is singular by partitioning the joint column space into orthogonal subspaces determined by the ratios of the generalized singular values, which reveal relative scalings between A and B. This partitioning enables robust numerical stability in computations, even when \beta_i = 0 for some i, avoiding direct inversion of B. The decomposition is instrumental in solving constrained least squares problems, such as minimizing \|Ax - b\|^2 subject to Bx = c, by transforming them into decoupled diagonal forms via the orthogonal factors, and in generalized eigenvalue problems. When B is the identity matrix, the GSVD reduces to the standard singular value decomposition of A, highlighting its role as a unifying framework that also supports generalized matrix inverses through the nonsingular X.[14]The GSVD was introduced by Charles Van Loan in 1976 as a generalization applicable to quotient eigenvalue problems and matrix pairs with shared columns, building on earlier ideas in numerical linear algebra.[13] Subsequent refinements by Christopher Paige and Michael Saunders in 1981 provided a more computationally efficient form, emphasizing direct algorithms like Jacobi rotations for stability.[14] In applications, the GSVD facilitates comparative data analysis, such as in bioinformatics for aligning and contrasting gene expression profiles across datasets by identifying shared and distinct patterns through the generalized singular value ratios, and in regularization of ill-posed problems.
Medicine
Spontaneous vaginal delivery
Spontaneous vaginal delivery (SVD) refers to the natural expulsion of a full-term infant through the vaginal canal in a vertex (head-first) presentation, without the use of operative interventions such as forceps, vacuum extraction, or cesarean section.[15] This process relies on the inherent forces of labor and is the most common mode of childbirth for uncomplicated pregnancies.[16]The procedure unfolds in three primary stages. The first stage involves cervical dilation from 0 to 10 cm, divided into latent (early) and active phases, accompanied by regular uterine contractions; for first-time mothers, this stage typically lasts 12 to 18 hours.[17] The second stage begins with full dilation and entails the mother's active pushing to deliver the baby, usually lasting 30 minutes to 2 hours.[18] The third stage concludes with the expulsion of the placenta, generally within 30 minutes post-delivery.[19]Clinical suitability for SVD requires a full-term pregnancy (37-42 weeks gestation), cephalic presentation, and absence of maternal or fetal complications such as preeclampsia, gestational diabetes, or signs of fetal distress.[15] Continuous monitoring of fetal heart rate and uterine contractions is essential throughout labor to ensure safety and detect any deviations prompting intervention.[20]SVD offers several advantages over operative deliveries, including a lower risk of maternal infection, shorter hospital stays, and faster postpartum recovery, which facilitates earlier initiation of breastfeeding and maternal-infant bonding.[16] In low-risk pregnancies, the success rate of achieving SVD approaches 70-80%.[21]Despite its benefits, SVD carries potential risks, including perineal tears graded from first-degree (superficial skin laceration) to fourth-degree (involving rectal mucosa and sphincter), which occur in up to 85% of vaginal births and may require suturing.[22] Other complications encompass postpartum hemorrhage due to uterine atony or lacerations, affecting about 5% of deliveries, and transient fetal distress from prolonged labor.[23]The term "spontaneous vaginal delivery" became standardized in 20th-century obstetrics as medical practices emphasized natural labor processes, particularly in contrast to the rising operative deliveries following the 1980s surge in cesarean section rates, which climbed from around 5% in 1970 to over 20% by the early 1990s in the United States.[24][25]As of 2023, with operative vaginal deliveries accounting for about 3% of births, SVD comprises approximately 65% of all births in the United States.[26][27]
Small vessel disease
Cerebral small vessel disease (CSVD), also known as small vessel disease (SVD) in the brain, refers to a spectrum of pathological changes affecting small penetrating arteries, arterioles, capillaries, venules, and small veins with diameters less than 500 μm, leading to chronic cerebral ischemia or hemorrhage.[28] It is characterized by structural and functional alterations in these vessels, often resulting from age-related degeneration or vascular risk factors, and manifests as neuroimaging abnormalities such as white matter hyperintensities, lacunar infarcts, cerebral microbleeds, and enlarged perivascular spaces.[29] CSVD is highly prevalent in older adults, with MRI markers present in 20–90% of individuals over age 65, depending on the specific lesion type and population studied; for instance, white matter hyperintensities affect up to 50% of those aged 70 and older.[30] It accounts for approximately 25% of ischemic strokes and 45% of vascular dementia cases worldwide.[31]The pathophysiology of CSVD involves multiple mechanisms, including arteriolosclerosis (such as lipohyalinosis and fibrinoid necrosis), microatheroma formation, and blood-brain barrier disruption, which collectively impair cerebral blood flow and cause tissue damage.[32] These changes lead to ischemic injury in deep brain structures and subcortical white matter, resulting in demyelination, axonal loss, and gliosis, visible as hyperintense lesions on T2-weighted MRI sequences.[33] Enlarged perivascular spaces and microbleeds further contribute to neurodegeneration, exacerbated by endothelial dysfunction and neuroinflammation.[29]Hypertension is the primary modifiable risk factor, present in over 70% of cases and promoting vessel wall thickening and luminal narrowing; diabetes, hyperlipidemia, and smoking also play significant roles.[34] Age greater than 65 years independently increases susceptibility, with prevalence of key markers like lacunar infarcts rising from approximately 5-8% in middle-aged adults (aged 50-60) to 20-30% in those over 70 years.[35][36]Clinically, CSVD is often silent in early stages but progresses to symptoms including cognitive impairment (particularly executive dysfunction), gait disturbances, mood disorders, and urinary incontinence, ultimately contributing to vascular dementia in advanced cases.[37] Diagnosis relies on multimodal MRI protocols to detect hallmark features: lacunar infarcts (small cavities <15 mm), confluent white matter hyperintensities (Fazekas scale ≥2), cerebral microbleeds (on susceptibility-weighted imaging), and enlarged perivascular spaces (>3 mm).[30] Assessment includes evaluating vascular risk factors, with hypertension confirmed in about 80% of symptomatic patients through ambulatory monitoring.[38] There is no curative treatment for CSVD; management focuses on prevention through rigorous blood pressure control (target <130/80 mmHg in hypertensives), statin therapy for dyslipidemia, and antiplatelet agents such as aspirin or clopidogrel to reduce recurrent stroke risk.[39] Lifestyle interventions, including smoking cessation and Mediterranean diet adherence, are recommended to mitigate progression.[40]The condition was first systematically described in the 1960s by neurologist C. Miller Fisher through autopsy studies, which linked lacunar infarcts to lipohyalinosis in hypertensive patients and established its distinct clinical syndromes.[41] Advances in neuroimaging, particularly MRI in the 1990s, revolutionized detection by enabling in vivo identification of subclinical lesions, shifting CSVD from a postmortem diagnosis to a clinically actionable entity.[42] As of 2025, ongoing research emphasizes early intervention in at-risk populations to curb its growing burden amid aging demographics.[43]
Weapons
SVD (rifle)
The SVD (Snayperskaya Vintovka Dragunova), or Dragunov Sniper Rifle, is a semi-automatic designated marksman rifle developed by Soviet designer Yevgeny Dragunov between 1958 and 1963.[44][45] It was created in response to the need for a squad-level precision weapon to counter NATO's adoption of intermediate cartridges, emphasizing rapid semi-automatic fire over long-range bolt-action sniping.[46] The rifle was selected after competitive trials against designs from other Soviet engineers and officially adopted by the Soviet Army in 1963 as the standard marksman rifle.[47]The SVD operates on a short-stroke gas piston system, chambered in the 7.62×54mmR cartridge, with a 10-round detachable box magazine.[48] It features a fixed wooden stock with an adjustable cheek rest, integral bipod, and a standard PSO-1 4× optical sight for precise targeting.[49] Key specifications include an overall length of 1,225 mm, unloaded weight of 4.3 kg (with optic), barrel length of 620 mm, muzzle velocity of 830 m/s, and a practical rate of fire of 30 rounds per minute.[48][50] The effective range is 800 m, with design accuracy of approximately 1.04 MOA vertical spread at 100 m using match-grade ammunition, though practical performance averages around 2 MOA depending on conditions and ammo quality.[51]Historically, the SVD entered mass production in 1964 at the Izhevsk Mechanical Plant and saw extensive combat use, including by North Vietnamese forces during the Vietnam War starting in 1968 and Soviet troops in the Afghan War from 1979 to 1989.[52][53] It has appeared in numerous modern conflicts worldwide, reflecting its role in Soviet and post-Soviet military doctrine as a designated marksman rifle rather than a true sniper weapon, prioritizing squad support fire at intermediate ranges over extreme precision.[45] The base SVD model features a fixed stock and serves as the foundation for variants, such as the folding-stock SVDS for compact applications.[48]As of 2025, the SVD remains in active service with Russian forces and has been exported to numerous countries, with ongoing production and modernized versions incorporating updated ergonomics and optics to extend its relevance.[54][55]
SVDS (rifle)
The SVDS (Snayperskaya Vintovka Dragunova Skladnaya), or Dragunov sniper rifle with folding stock, is a compact variant of the SVD developed in the early 1990s by Izhmash for enhanced mobility in specialized units.[56] Introduced in 1995 primarily for airborne and special forces, it features a side-folding polymer stock that significantly reduces the rifle's length when collapsed, improving transportability for parachute drops, vehicle operations, and urban environments while preserving the core gas-operated, semi-automatic mechanism of its predecessor.[57][56] The design retains the 7.62×54mmR caliber for consistent ballistics and uses a 10-round detachable steel magazine, with an effective firing range of around 800 meters.[56]Key modifications include a shortened barrel measuring 565 mm—compared to 620 mm on the baseline model—to balance compactness and velocity, along with a separate pistol grip and adjustable cheekpiece for better ergonomics during prolonged use.[57][56] The rifle's side-mounted rail system supports the addition of Picatinny adapters for contemporary optics, suppressors, and night vision devices, enhancing versatility without compromising the inherent accuracy suited for designated marksman roles.[57]Standard specifications for the SVDS include an overall length of 1,135 mm unfolded and 875 mm folded, with an unloaded weight of 3.7 kg (approximately 4.7 kg including optics and empty magazine).[56][57] These attributes enable effective engagement of personnel targets at medium ranges while prioritizing rapid deployment in dynamic scenarios, such as urban combat or mounted assaults. Production at Izhmash—now part of the Kalashnikov Concern—began in 1991 and remains active as of 2025, with output significantly increased to fulfill defense contracts amid ongoing demand.[54][56]The SVDS serves an operational role emphasizing portability over the fixed-stock SVD, supporting squad-level precision fire in high-mobility operations while maintaining sub-MOA accuracy with match-grade ammunition at 500–600 meters.[57] Known variants include the standard SVDS and the further upgraded SVDM, which incorporates an adjustable folding stock, improved rail integration, and enhanced ergonomics; the SVDM was presented in 2015 and entered serial production shortly thereafter.[58][59]
Other uses
Society of the Divine Word
The Society of the Divine Word (Societas Verbi Divini, SVD) is a clerical male religious institute of the Catholic Church, founded on September 8, 1875, by Arnold Janssen, a German priest, in Steyl, Netherlands, amid the Kulturkampf anti-Catholic policies in Germany.[60] Janssen established the congregation to train missionaries for evangelization in non-Christian lands, emphasizing the proclamation of the Gospel through preaching, education, and emerging media forms. Members profess the traditional religious vows of poverty, chastity, and obedience, along with an oath of dedication to the missions, committing them to lifelong service in cross-cultural contexts.[61] The society's mission centers on making the Word of God known worldwide, particularly among marginalized and indigenous peoples, fostering inculturation by integrating Christian faith with local cultures through studies in linguistics and anthropology.[62]The congregation received pontifical approval from the Holy See in 1905, enabling its rapid expansion beyond Europe.[63] Early missionaries were sent to China in 1879, marking the first foreign venture, followed by establishments in Africa (Togo, 1892), the Americas (United States, 1895), and Asia (Philippines, 1909).[61] By the 1980s, membership peaked at over 6,000 priests and brothers, reflecting global growth; as of 2025, it maintains 5,754 members.[64] Founder Arnold Janssen, beatified in 1975, was canonized as a saint by Pope John Paul II on October 5, 2003, alongside Joseph Freinademetz, recognizing their pioneering missionary zeal.[65] SVD members also contributed to the Second Vatican Council's documents on missions, particularly Ad Gentes, advocating for culturally sensitive evangelization and interreligious dialogue.[66]Organizationally, the SVD is divided into over 60 provinces and regions across six zones, with a focus on formation in linguistics and anthropology to support inculturation efforts, such as adapting liturgy and catechesis to local traditions.[67] The society operates universities, high schools, and seminaries worldwide, alongside media apostolates including radio and television stations like Radio Veritas Asia, which broadcasts in multiple languages to promote faith and social justice.[68] Key activities encompass Bible translation projects in numerous indigenous languages, disaster relief operations, and initiatives for justice, peace, and interreligious harmony, often in partnership with lay associates.[69]Today, the SVD is active in more than 70 countries, with significant presence and focus in Papua New Guinea, India, and the Philippines, where members engage in parish ministry, education, and advocacy for indigenous rights.[64] This religious congregation is distinct from secular media entities sharing the SVD acronym, such as the Swedish newspaper Svenska Dagbladet.[70]
Svenska Dagbladet
Svenska Dagbladet, often abbreviated as SvD and translating to "Swedish Daily Paper," is a prominent national newspaper based in Stockholm, Sweden. Founded on December 18, 1884, it was established as a conservative publication aimed at providing a voice against emerging socialist movements in Swedish society.[71] The newspaper has maintained a daily broadsheet print format alongside a robust online edition, adapting to digital transformation while preserving its role as a key source for national news.[72] Ownership transferred to the Norwegian media conglomerate Schibsted in 1998, when Schibsted acquired a 73.6% stake, later increasing to full control, solidifying its position within a major Nordic media group.[73]Historically, Svenska Dagbladet began as an anti-socialist outlet during its early years, reflecting the political tensions of late 19th-century Sweden. Under editor-in-chief Ivar Andersson from 1940, it adopted a firm anti-Nazi and continued anti-socialist stance, navigating World War II-era challenges.[71] By the 1970s, the paper shifted toward greater independence, formally declaring itself unaffiliated with political parties in 1977 after decades of ties to conservative groups like the General Electoral League.[74] The 1990s marked a pivotal digital pivot, with early adoption of online publishing, and in 2000, it transitioned to a tabloid format to enhance readability and accessibility.[75] A landmark achievement came in 2005 when Svenska Dagbladet won the European Newspaper Award for best nationwide newspaper, recognizing its design and journalistic excellence.[76]The newspaper's editorial stance is described as independently moderate or liberal-conservative, positioning it as center-right in Sweden's media landscape, with a focus on balanced coverage of politics, business, and culture.[74] It is renowned for investigative journalism, including significant contributions to the 2016 Panama Papers revelations, where SvD reported on hundreds of Swedish individuals and firms implicated in offshore financial structures, prompting scrutiny of domestic banks.[77] Key sections include Nyheter for general news, Kultur for arts and cultural analysis, Sport for athletics coverage, and business-oriented supplements like Placeringsguiden, which provides financial guidance.[72] In the 1970s, SvD played a role in exposing political scandals through its reporting on events like the Geijer affair, involving high-level government figures and organized prostitution, contributing to broader public discourse on corruption.[78]In recent years, Svenska Dagbladet has adapted to evolving media dynamics by implementing a strengthened paywall model around 2017, emphasizing premium content to drive subscriptions.[75] As of 2025, its print circulation hovers around 100,000 copies daily, while digital subscribers exceed 125,000, reflecting a historic surge in online engagement amid declining traditional print readership.[79] The paper maintains influence through focused reporting on sustainability and AI ethics, addressing contemporary challenges like climate change science and the ethical implications of artificial intelligence in society, aligning with Schibsted's broader commitment to responsible media practices.[80][81]