Singular Value Decomposition: LAPACK Computational Routines

Developer Reference for Intel® oneAPI Math Kernel Library for C

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Date 4/28/2026

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Singular Value Decomposition: LAPACK Computational Routines

This topic describes LAPACK routines for computing the singular value decomposition (SVD) of a general m -by- n matrix A :

A = UΣV^{H} .

In this decomposition, U and V are unitary (for complex A ) or orthogonal (for real A ); Σ is an m -by- n diagonal matrix with real diagonal elements σ _i :

σ_{1}≥σ_{2}≥ ... ≥σ_{min(m, n)}≥ 0 .

The diagonal elements σ _i are singular values of A . The first min(m, n) columns of the matrices U and V are, respectively, left and right singular vectors of A . The singular values and singular vectors satisfy

Av_{i} = σ_{i}u_{i} and A^{H}u_{i} = σ_{i}v_{i}

where u_i and v_i are the i -th columns of U and V , respectively.

To find the SVD of a general matrix A , call the LAPACK routine ?gebrd or ?gbbrd for reducing A to a bidiagonal matrix B by a unitary (orthogonal) transformation: A = QBP^{H} . Then call ?bdsqr , which forms the SVD of a bidiagonal matrix: B = U_{1}ΣV_{1}^{H} .

Thus, the sought-for SVD of A is given by A = UΣV^{H} =(QU_{1})Σ(V_{1}^{H}P^{H}) .

Computational Routines for Singular Value Decomposition (SVD)

Operation	Real matrices	Complex matrices
Reduce `A` to a bidiagonal matrix `B` : `A = QBP^{H}` (full storage)	?gebrd	?gebrd
Reduce `A` to a bidiagonal matrix `B` : `A = QBP^{H}` (band storage)	?gbbrd	?gbbrd
Generate the orthogonal (unitary) matrix `Q` or P	?orgbr	?ungbr
Apply the orthogonal (unitary) matrix `Q` or P	?ormbr	?unmbr
Form singular value decomposition of the bidiagonal matrix `B` : `B = UΣV^{H}`	?bdsqr ?bdsdc	?bdsqr

You can use the SVD to find a minimum-norm solution to a (possibly) rank-deficient least squares problem of minimizing ||Ax - b||_{2} . The effective rank k of the matrix A can be determined as the number of singular values which exceed a suitable threshold. The minimum-norm solution is

x = V_{k}(Σ_{k})^{-1}c

where Σ _k is the leading k -by- k submatrix of Σ , the matrix V_k consists of the first k columns of V = PV_{1} , and the vector c consists of the first k elements of U^{H}b = U_{1}^{H}Q^{H}b .

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Developer Reference for Intel® oneAPI Math Kernel Library for C

Singular Value Decomposition: LAPACK Computational Routines